Advanced SAT Type Questions Flashcards

Question

If you decide to backsolve, which answer choice do you plug in first?

Answer 1

Start with the *middle* answer choice "C". In some cases work backwards from choice "E". For example, when you are asked to find the largest possible value.

Answer 2

To avoid "traps": * improve your core knowledge * read slowly * solve problems to the end Don't stop half-way and try not to semi-guess. Arrive at your own answer, *then* compare it to the choices available.

Answer 3

You *cannot* determine which of the five numbers in the problem is the smallest with the information given. ## Footnote If *k* is a positive integer, then the smallest number is *k*. But *not all numbers are positive integers*! *k* can be negative. In this case, *k* is the greatest number, not the smallest.

Answer 4

When you pick numbers to substitute for variables, make sure to test *all possible* choices. *Example*: If the problem states that *n* \> 0, *n* can be a whole number as well as a fraction. If it says that n is an integer, don't forget to test negative numbers.

Answer 5

of tickets distributed = Last ticket # - First ticket # + 1 * x* - 29 + 1 = 254 ⇒ *x* = 254 + 29 -1 ⇒ * x* = 282 If your answer was 283, you added 29 and 254 but didn't subtract 1. You've fallen into a trap the test makers prepared for you. \*\*\* Change the numbers to small numbers (like 2 and 10) and count.

Answer 6

To count *consecutive* integers in a range, subtract the smaller integer from the larger and then *add 1* to the difference. *Example:* How many consecutive integers are between 5 and 10 inclusively? Let's count. 5, 6, 7, 8, 9, and 10. There are six integers between 5 and 10. 10 - 5 + 1 = 6

Answer 7

Make sure you answer the question that is being asked. ## Footnote Yes, you could have solved the problem correctly but didn't read the instructions carefully. The last step in the successful solution process is to evaluate the answer.

Answer 8

1. The team didnt necessarily play 20 games this year. It may have played 20, 40, ... or any multiple of 20 games this year. 2. Subtracting from or adding numbers to the ratio *will change* the ratio. A 7 to 13 ratio doesn't equal a 2 to 8 ratio. 3. Multiplying or dividing both parts of the ratio by the same number *will not change* the ratio. A 7 to 13 ratio equals a 35 to 65 ratio.

Answer 9

The common part (blonde students) of two ratios is not the same number. The numbers in these ratios are *not proportional* because the ratios don't refer to the same whole. * Blondes* : Reds = *9* : 2 * Blondes* : Brunettes = *3* : 5 Re-state the ratios so that the common part is represented by the *same* number. The least common multiple of 3 and 9 is 9. Re-state 2nd ratio. *Blondes* : Brunettes = *9* : 15. Only now you can compare the ratios. But... make sure you are answering the question! You are asked for the ratio of brunettes to red heads, so it's 15 : 2.

Answer 10

Students are often confused about these simple concepts. * Ratios may *not* represent actual quantities. * You *cannot* add or subtract from a ratio or part of the ratio. * When common parts of two ratios don't refer to the same whole, the numbers in the ratios are *not proportional*. \*\*\* Remember, ratios are a lot like fractions. The same rules apply.

Answer 11

**Avg Rate = ^{Total # of pages} */* _{Total time}** ## Footnote We know that the book has *2n* pages. We don't know the time it took you to read each part but we can write equations to find it: time ₁ = *ⁿ/*₁₀ (*n* pages at 10ppd rate) time ₂ = *ⁿ/*₃₀ (*n* pages at 30 ppd rate) To make it easier, pick a number for *n* that is a multiple of both 10 and 30. Or solve it algebraically. Total time = time ₁ + time ₂ = *ⁿ/*₁₀ + *ⁿ/*₃₀ = ^{3n + n}*/*₃₀ = ⁴ⁿ*/*₃₀ Avg Rate = *2n* ÷ *⁴ⁿ/*₃₀ = *2n* \* ³⁰*/*_4n = 15 The correct answer is choice (b) 15.

Answer 12

Taking the average of 10 and 30 to find the average rate of reading is wrong! It doesn't take into account that it took you more time to read the first part of the book because your average rate of reading was slower. You can pick numbers to solve this problem instead of writing equations. The LCM of 10 and 30 is 30. Let's set *n* as 30 pages. There are 60 pages in the entire book. It took you 3 days (30 ÷ 10) to read the 1st part and 1 day (30 ÷ 30) to read the 2nd part. So, you read 60 pages in 4 days. Your average rate of reading for the entire book is 15 pages/day.

Answer 13

Many students forget that the *average rate of A per B is the total of A divided by the total of B.* For example, the average rate of speed of the whole trip is *not* the average of the rates of speed on different legs of the trip. Avg Rate of Speed = ^{Total Distance}*/*_{Total Time}

Answer 14

You can solve the problem algebraically but let's practice the "pick numbers" strategy. Suppose the original population of town A was 100. In 5 years, it became 110 (100 \* 1.1). Over the next 5 years it increased upto 132 (110 \* 1.2). The change in population is 32, the original population is 100. Therefore, the population increased by 32%, choice (b). If you picked choice (a) 30%, you've added 20% to 10% and got caught in a trap.

Answer 15

* Even without calculating exact amounts you can determine that the increase is smaller than the decrease. The reason is that the 25% increase is calculated off of 1,000 while the 25% decrease is taken off of a greater amount. * The value of your portfolio after two years is smaller than the original value. 1st year: 1,000 x 1.25 = 1,250 2nd year: 1,250 x 0.75 = 937.50

Answer 16

1st year: 1,000 x 1.25 = 1,250 2nd year: 1,250 x 0.75 = 937.5 ^937.5/₁₀₀₀ \* 100% = 93.75%, choice (e). Answer choices (a) and (c) are traps. If you picked (a), you didn't read the problem carefully and thought that both years the value increased by 25%. Then, you added up percent increases. If you picked (c), you mistakenly assumed that the amount of the increase equals the amount of the decrease.

Answer 17

* *Multiple percent changes are not additive*. A 20% decrease and an additional 60% decrease off the original price do *not* equate to an 80% decrease. Percentages can be added or subtracted only when they are calculated from the same amount. * The *same percent* increases or decreases *don't* necessarily *equal* the *same amounts* of decrease of increase.

Answer 18

Guessing is smart as long as you are using "educated" guessing. * Educated guesses are guesses that are made based on a familiarity with the math concepts in the question being asked * Advanced students should make educated guesses whenever they can eliminate one answer choice * Intermediate students should make educated guesses only when they can eliminate two answer choices * Always guess on grid-ins because points are not lost for wrong answers!

Answer 19

Math questions are arranged in order of difficulty - *easy to hard*. On the average, you can spend *60 to 90 seconds per question*. But, with practice, you will be able to solve easier questions in the beginning of each section faster, leaving more time for harder ones. * In a section that gives you 25 minutes for 20 questions, try to get those first 10 easy questions solved in about 7 minutes. * Try to have 2 mins per question towards the end of the section, on the last 3 or 4 problems.

Answer 20

It's very important that you manage your time wisely during the test. * *Don't get stuck on a question*. If you see that it's taking you too long, move on. If time allows, you can come back to it * *All questions are worth the same points*. Make sure you answer all easy questions (usually 1 through 10) in each section * *Keep track of time*. To get the feel of how much time you are spending per question, take a lot of practice tests as if they were the real tests

Answer 21

To solve remainder problems, you can use this formula to figure out any missing element. ***N = Q x D + R*** * *N* - Number in question (or Dividend) * *Q* - quotient * *D* - divisor * *R* - remainder *Example*: When *N* is divided by 5, the remainder is 4. What are the first three values of *N*? *N* = *Q* x 5 + 4 The first three values of *N* are 9 (when *Q* = 1), 14 (when *Q* = 2), and 19 (when *Q* = 3).

Answer 22

* _The purpose_ of the problem is to divide certain numbers to find a remainder * _The question_ is to find a *remainder* of a certain number when *divided by 5* * _Extract key info:_ when a divisor of a certain number is 5, the remainder equals 3 * _Identify formula/strategy:_ ***N = Q x D + R*** (*N* - number in question, *Q* - quotient, *D* - divisor, *R* - remainder) * _Apply all knowledge_ to solve: If *N* = *Q* x 5 + 3 ⇒ 2N = 2 (5Q + 3) = 10Q + 6 10Q is divisible by 5. When 6 is divided by 5, the remainder is 1. * Make sure your _answer_ _makes sense_ and is the _answer to the question_.

Answer 23

Once you've analyzed the information, set up the formula ***N = Q x D + R*** and plug in what you know. (*N* - number in question, *Q* - quotient, *D* - divisor, *R* - remainder). *N* = *Q* x 6 + 4. Possible choices for *N* are 10 (*Q* = 1), 16 (*Q* = 2), 22 (*Q* = 3), etc. Examining answer choices, you will see that adding 8 to those numbers will leave no remainder when divided by 6. The correct answer is (b) *N* + 8. Another way to solve it is to use logic. A zero remainder when divided by 6 means that we are looking for multiples of 6; i.e. 6, 12, 18, ... What number should you add to 4 to make it one of those multiples? 2, 8, 16, ... N + 8 is choice (b).

Answer 24

The essence of this divisibility problem is that if a certain number is divided by 6, the remainder is 4. When that number is reduced by 2 and divided by 8, there is no remainder. Let's set up the remainder formula. ***N = Q x D + R*** Realize that *Q* is, in this case, the number of *students*, *D* is the number of *pens* given to each. * N* (# of pens) = *S* (students) x 6 + 4 * N* (# of pens) = *S* - 2 (students) x 8 6S + 4 = (S - 2) x 8 ⇒ 6S + 4 = 8S - 16 Solving for S, you get that there are 10 students in the class originally. When 2 students are absent, there are 8 students left in the class. 8 x 8 = 64 - # of pens before distribution.

Answer 25

Think of purpose, define the question, underline key info, set up the remainder formula, and plug in what you know: ***N = Q x D + R*** In this case, *N* = 18, *R* = 3 18 = *Q x D* + 3 ⇒ *Q x D* = 15 There are three pairs of integers whose product is 15. (1; 15), (3; 5), and (5; 3). So, there are *three* positive integers that make the statement true. Logically, 18 - 3 = 15. So, we need to find positive integers that are factors of 15. They are 1, 3, 5, and 15 but only *3, 5, and 15* work for this problem. 18 divided by 1 doesn't leave a remainder of 3.

Answer 26

You can *pick small numbers* to substitue for variables. Pick *n* = 6 and *m* = 10, plug into the answer choices and determine which ones are divisible by 15. The correct answer is choice *i and iii*. You can solve this problem logically or algebraically as well. The number is divisible by 15 if it's divisible by *both* 3 and 5. *n* is a multiple of 3 and can be expressed as 3*x*. *m* is a multiple of 5, so you can write it as 5*y*. *n* = 3*x*; *m* = 5*y*. * *n \* m* = 3*x* \* 5*y* = 15*xy* - yes * 5 (*m* + *n*) = 5 (3*x* + 5*y*) = 15*x* + 25*y* - not always * 5*n* + 3*m* = 5\* 3*x* + 3\* 5*y* - yes

Answer 27

* _Purpose of the problem:_ Intersection of sets, division operation * _Question:_ Find *N* that meets all criteria * _Key info:_ 38 \< *N* \< 88 When *D* = 5, *R* = 0. When *D* = 4, *R* = 2. When *D* = 6, *R* = 2 * _Formulas/Strategy:_ Find the intersection of 3 sets using just logic. * _Apply all knowledge and solve:_ List all multiples of 5 between 38 and 88. {40, 45, 50, 55, 60, 65, 70, 75, 80, 85}. The number that is divisible by both 4 and 6 is a multiple of 12. List all multiples of 12 in a target range. {48, 60, 72, 84} But our *N* leaves a remainder of 2 when divided by 4 or 6 so *N* has to be *2 greater* that listed multiple of 12. The only number that's in both sets is 50. * _Make sure you answered the question._ Yes, *N* = 50

Answer 28

**Average (A)** = **^{Sum of the terms (S)}*/*_{Number of terms (N)}** and therefore, ** S****um = Avg x N** In the next few cards, we will show you how to use these basic formulas to solve different types of average problems.

Answer 29

When you divide the sum of numbers in a set by the average of those numbers, you find how many of those numbers are in the set.

Answer 30

* The _purpose_ and the _question_ is to find the average of five positive integers number * _Extract key info:_ The median of five positive integers is 36. The difference between the smallest and the largest is 4 * _Make logical conclusions_: These five integers must be consecutive since the difference between the smallest and the largest integers is 4 * _Identify formulas and strategy_: In any set of *evenly spaced* numbers, the *median* number is also the *average.* The answer is 36, choice (c) * _Is it the answer to the question?_ Yes.

Answer 31

* _The purpose_ is to find the missing term in a set using the average of the set. * _The question_ is to find the price of the added CD. * _Extract key info:_ the average price of 5 CDs is $12, the average price of 6 CDs is $13. * _Make logical conclusions:_ if you know the average and the number of terms, you can find the sum. * _Identify math formulas and strategies_: **Sum (S)** ÷ **Number of terms (N)** = **Avg** ⇒ **S = Avg x N** * _Apply all your knowledge to solve_: Total cost of 5 CDs: 5 x 12 = 60 Total cost of 6 CDs: 6 x 13 = 78 78 - 60 = $18 (price of the added CD) * _Make sure the answer answers the question:_ yes.

Answer 32

* _The purpose_ is to use the average to find the sum of terms * _The question_ is to express *a +* *b* in terms *c, d,* and *f* * _Extract key info:_ The average of 4 terms *a, b, c* and *d* is *f* * _Identify math formulas and strategies:_ You can use simple, small numbers to replace variables. Or solve it algebraically **Sum = Avg \* Number of Terms** * _Apply all your knowledge to solve:_ (*a + b + c + d*) ÷ 4 = *f* ⇒ * a + b + c + d* = 4*f* ⇒ * a + b* = 4*f - c - d* .... choice (c) * _Is it the answer to the question?_ Yes

Answer 33

Analyze the purpose, the question and the key info on your own. * _Identify math formulas and strategies:_ **Sum = Avg \* Number of Terms** * _Logical conclusion:_ given the average of two numbers you can find their sum. * _Apply all knowledge to solve:_ *x + y* = 60 *x + z* = 90 Subtract 1st equation from the 2nd equation. * z - y* = 30 * _Make sure you answered what was asked_. Yes

Answer 34

You should analyze the purpose, the question, key information and formulas and strategies that may apply here. **Sum = Avg x Number of Terms** * _Key info_ you are given: group 1: *X* ppl, 140 lbs ave group 2: *Y* ppl, 128 lbs ave combined: *X + Y* ppl, 130 lbs ave * _Use the formula_ above to find the sum of weights in each group of people: Sum of weights group 1 = *X* \* 140 Sum of weights group 2 = *Y* \* 128 Combined groups' weight = (*X + Y*) \* 130 Set up an equation: *X* \* 140 + *Y* \* 128 = (*X + Y*) \* 130 ⇒ 140*X* + 128*Y* = 130*X* + 130*Y* 10*X* = 2*Y* ⇒ ***X : Y = 2 : 10***

Answer 35

Again, you have to mathematically connect the average, the number of terms, and the sum of terms. The formula that ties all three together is **Sum = Avg x Number of Terms** The key information here is that the average of seven positive numbers is 9. You need to find the greatest of them. Sum = 7 x 9 = 63 One term is the *greatest* possible when other terms in a set are the *smallest* possible. The smallest positive integer is 1. So, 6 out of 7 people could have written 1. 63 - 6 x 1 = 57 The greatest possible integer is 57.

Answer 36

Your general approach should be the step-by-step logical process taught in this deck. A must know core knowledge is the average formula. The best strategy is to set up the formula and solve the problem algebraically. **Sum = Avg x Number of Terms** Let's call the first two numbers *a* and *b*. Their average is w. So, (*a + b*) ÷ 2 = *w* ⇒ *a + b* = 2*w* One more number *p* is added to the set. The sum of three numbers is 2*w + p*. The average of three numbers is: Avg = (2*w* + *p*) ÷ 3

Answer 37

By definition of the average, if the sum of six integers is 60, their average is 10. The average of a set of evenly spaced numbers is also the median of the set, so the median is 10 as well. In a set with even number of terms, the median is the average of the two middle values, 9 and 11. {5, 7, 9, 11, 13, 15} Or solving algebraically... *n* + (*n* + 2) + (*n* + 4) + (*n* + 6) + (*n* + 8) + (*n* + 10) = 60 6*n* + 30 = 60 ⇒ 6*n* = 30 ⇒ *n* = 5 If we add 60, we are adding the 7th term. The median becomes 11. {5, 7, 9, *11*, 13, 15, 60}

Answer 38

In this problem, concetrate your efforts on analyzing answer choices. Picking numbers works as a strategy and there are no restrictions to the numbers you pick. For example, the set is {2, 6, 8, 14, 15}. The median is the middle value of the set. Review each answer choice. Increasing the smallest number (a) or decreasing the largest number (e) may or may not change the median. Choices (b) and (c) will definitely change the median. The only operation that will *never* change the median value of a set is increasing the largest number, choice (d).

Answer 39

A *union* of sets is all elements in those sets without repeated elements. An *intersection* of sets is only common elements that are in all sets. An *empty* set is a set with no elements.

Answer 40

The set that contains absolute values has all positive integers. Since the first set contains all negative integers and the second set contains all positive integers, their *intersection* is *an empty set*. You have to know the definitions of an *intersection* of sets and an *empty* set. The intersection of two sets is a set that includes only *common* elements in both sets. An empty set has *zero* elements.

Answer 41

If you know the definition of a union of sets, you can easily solve this problem. The *union* of two sets is *all* elements in both sets without *repeating*. Write out the elements of both sets: * X* = {5, 7, 11, 13, 17, 19} * X* U *Y* = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19} Set Y must be {1, 3, 9, 15} The minimum number of elements in set *Y* is 4, choice (a).

Answer 42

The strategy is to understand the concept of intersections of sets really well so you can work backwards. The *intersection* of two sets consists of only elements that are in *both* sets. The intersection set consists of 5, 12 and 16. Therefore, these numbers have to be in both set *A* and set *B*. 12 is in both sets but 5 and 16 are only in set *A*. By definition, 5 and 16 have to be in set *B* as well. The missing elements of set *B* are 5 and 16.

Answer 43

An *intersection* set consists of elements *common* to all three sets. To figure out the first five elements of the intersection set you can list a certain number of elements of each of the three sets and find the common elements. A much faster way is to realize that you need to find the least common multiple of 12, 20 and 15. The LCM of these numbers is 60 (3 x 4 x 5). Therefore, the intersection set is a set of multiples of 60. The first five elements in the intersection of *A*, *B*, and *C* are {60, 120, 180, 240, 300}.

Answer 44

By definition, the union of A, B, and C is *all* elements of A, B, and C without repetition. The intersection of A, B, and C is the *common* elements of the three sets. The union of A, B, and C is the set of *all* multiples of 12, 20, and 15. It gives you the *most* number of elements. The intersection of A, B, and C is the set of multiples of 60 (the LCM of 12, 20, and 15). It gives you the *least* number of elements. Choice (c) is correct.

Answer 45

The sequences (or progressions) you need to know for the SAT are *arithmetic* and *geometric*. The difference between them is that in an arithmetic sequence there is a constant *difference* between two consecutive terms; in a geometric sequence there is a constant *ratio* between two consecutive terms. _Arithmetic_: a_n = a₁ + d \* (n - 1) _Geometric_: a_n = a₁ \* r^{(n - 1)}

Answer 46

# * _State the purpose:_ Determine the pattern and terms of a sequence with simple arithmetic operations * _Define the question:_ Find the 20th term * _Extract key info:_ You know the first 3 terms of an arithmetic sequence * _Make logical conclusions from the info given:_ You can find the common difference * _Identify math formulas and strategies that may apply:_ Core knowledge such as the definition of an *arithmetic* sequence, the *common difference* *formula* and the *formula for the nth term* An arithmetic sequence is defined by a common difference (*d*) between the terms. *d = a₂ - a₁* = 298 - 302 = - 4 *n*th term formula: ***a_n = a₁ + d \* (n - 1)*** a₂₀ = 302 + (- 4) \* (20 - 1) = 226

Answer 47

You can pick small numbers to make up an arithmetic sequence. There is a constant difference between any two consecutive terms. {3, 9, 15, 21, 27, 33} The mode is the number that appears most often. The median is the middle value of a set of numbers in order. The arithmetic mean is the average. Let's analyze the answer choices. * (i) *false* since there is no mode * (ii) *true* since the median is 18 ((15 + 21) ÷ 2). The mean is also 18 (108 ÷ 6) * (iii) *true* (3 + 33) ÷ 2 = 18 \*\*\* In an arithmetic sequence, the terms are *evenly* spaced. Therefore, the *median* is also the *average* of the set. And the average can be figured out by taking the average of the smallest and the largest term.

Answer 48

* _The purpose:_ To write a geometric sequence using multiplication * _The question:_ Find how many terms of the first 21 are \< 21 * _Extract key info:_ It's a geometric sequence. The first term *a₁* = 1. The common ratio *r* = - 2 * _Make logical conclusions:_ Write the sequence with info given and realize that negative and positive numbers alternate * _Determine formulas and strategy:_ Core knowledge and logic *a₂ = a₁ \* r , a₃ = a₂ \* r, .....* 1, -2, 4, -8, 16, -32, 64, ..... * _Apply all you know and solve_: 2nd, 4th, 6th, ... terms (i.e. even numbered) are negative so they all are less than 21. There are 10 of them among 21 terms. The 1st, 3rd, and 5th terms of the sequence are less than 21. * 13 terms of the first 21 are less than 21* * Check the question again

Answer 49

Analyze the purpose, the question, the key information, make a logical conclusion, determine the formulas and the strategy to use on your own. If the average of the first three terms is 13, their sum is 39. 39 - 3 - 27 = 9. *a₁* = 27, *a₂* = 9, *a₃* = 3, ... Common ratio *r* = ¹/₃ ⇒ *a₅* = ¹/₃. Found the answer? No! You were asked to find one-half of a₅. ¹/₃ \* ¹/₂ = ¹/₆

Answer 50

Your strategy here is to follow the steps of the critical thinking process and have an excellent knowledge of sequences. Since this is an arithmetic sequence, there is a common difference (*d*) between its consecutive terms. You should know the formula for the *n*th term. * *d* = *a₂ - a₁* = 18 - 11 = 7 * **a_n = a₁ + *d* (*n* - 1)** Plug in.... a_n = 11 + 7(*n* - 1) The trick here is to realize that the original expression in the problem should be distributed to look like the formula. a_n = 11 + 7 (25 - 2) = 11 + 7 \* 23 *n* - 1 = 23 ⇒ *n* = 24, choice (d).

Answer 51

* According to the formula below, to find the sum, you need to know the 1st and the 20th terms **Sum_n = ¹/₂*n* (*a₁ + a_n*)** * The formula for the *n*th term in an arithmetic sequence requires you to find *d* (common difference between consecutive terms) ***a_n = a₁ + d \*(n - 1)*** * a₆* = *a₁* + 5 \* *d* ; *a₂* = *a₁* + *d* * a₁* = *a₆* - 5 \* *d* = *a₂ - d* * a₆ - a₂* = 11 - 3 = 4*d* ⇒ *d* = 2 * Next, find the 1st and the 20th terms * a₁* = 3 - 2 = 1 * a₂₀* = 1 + 2(20 - 1) = 39 * Now you know all you need to figure out the sum: * S₂₀* = 1/2 \* 20(1 + 39) = 400

Answer 52

* _The purpose:_ To find the missing terms of a geometric sequence. * _The question:_ Find the value of *b* * _Key info:_ You know four terms of the geometric sequence: *a,* 4*,* 16*b, ab* * _Logical conclusion:_ In a geometric sequence, there is a common *ratio* between consecutive terms. * _Math formulas and strategy:_ Set up proportions and cross-multiply to solve 4 : *a* = 16*b* : 4 = *ab* : 16*b* 16 = 16*ab* and 16²*b*² = 4*ab* ⇒ *ab* = 1 and 64*b* = *a* ⇒ 64*b*²= 1 ⇒ *b* = ¹/₈ * _Does the answer make sense?_ Plug in *a* and *b* to double check

Answer 53

Core knowledge is a must. If the following problems present difficulties, go back to the "Percent, Ratios and Proportions" deck. All percent problems can be solved easily algebraically. In our view, setting up a proportion to find the unknown is the best way to approach more advanced questions. You can also pick numbers to substitute for variables or when figuring out a part of the unknown whole.

Answer 54

# * _State the purpose:_ Divide numbers to find a ratio * _Define the question:_ Find the ratio of A to the total T in percentages * _Extract key info:_ A is 75% of B, C is 50% of B * _Make logical conclusions_: A and C are compared to B so B is 100% in this case. Total number of students equals 75% + 100% + 50% = 225% of B * _Identify the strategy:_ You can solve the problem algebraically or pick a value for the number of students in class B * _Apply all knowledge:_ ^A*/*_T = ^{75% of B}*/*_{225% of B}= 0.33 or 33.3% * _Make sure your answer is the answer to the question_. Yes.

Answer 55

You know that 30% of an original price is $24. Conclude that 100% is the original price of the item and set up a proportion. You can write the proportion 2 ways: ³⁰*/*₁₀₀= ²⁴*/_x* or ²⁴*/*₃₀ = *^x/*₁₀₀ Cross-multiply to solve for *x. x* = 80 Is choice (d) the correct answer? No. It's a trap. You found the original price. The question asks for half of the original price. So, it's 40, choice (a) .

Answer 56

* _The purpose:_ Find a percent decrease or increase * _The question:_ Find change in the area in % * _Extract key info:_ Length increases by 50% while width decreases by 50% * _Make logical conclusions:_ The new length is 150% of the original or 1.5L. The new width is 50% of the original or 0.5W * _Identify formulas and the strategy:_ Know the percent change formula. Pick simple numbers for length and width or solve it algebraically **Percent change =** **^{Change in value}*/*_{Old value} x 100%** * _Apply all knowledge and solve:_ Let's assume L = 10, W = 20 so the original area is 200. New L = 15, new W = 10 so the area is now 150. 200 - 150 = 50 (the amount of change) 50/200 x 100% = 25% The area decreased by 25%, choice (e)

Answer 57

**Percent change =** **^{Change in value}*/*_{Old value} x 100%** The original area: *L x W* The new area: 1.5*L* x 0.5*W* = 0.75 *L x W* The difference between them is 0.25. Therefore, you can make the conclusion that the area decreased by 25%, choice (e).

Answer 58

Analyze the problem the way we recommend to approach all SAT questions. You can solve it algebraically or pick numbers for *n, m* and *p*. **Decreased value =** **(100 - % decrease) \* original value** Let *n* be $50, *m* - 10% and *p* - $20. Now, figure out the cost after two reductions (50 - 10% of 50) - 20 = (50 - ¹⁰/₁₀₀\* 50) - 20 ...... or (*n - ^m*/₁₀₀\* *n*) - p = ((1 - ¹/₁₀₀*m*) *n*) - *p* Realize that you needed to distribute to make the expression look like one of the answer choices. Choice (c) is the correct answer.

Answer 59

This is a long-winded word problem. We suggest that you use the active reading strategy for these types of questions. When working with paper SAT materials, highlight and/or underline key information. Re-tell the story described in the problem. Diagram the information and clearly label it. If you are an intermediate student*, do not* go straight to setting up an equation. * x* - original price ; 0.75*x* - John's cost 0. 75*x* \* 1.25 - Mark's cost (1. 25 \* 0.75x + 200) \* 1.08 - Jim's cost Now, let's solve for *x*. (1. 25 \* 0.75x + 200) \* 1.08 = 1,377 0. 94 *x* \* 1.08 + 216 = 1,377 ⇒ *x* = 1,139

Answer 60

The problems on Numbers Theory and Operations primarily test your knowledge of properties of the integers such as *prime* and *composite* numbers, *odd* and *even* numbers, *positive* and *negative* numbers. You should know how to find the *LCM*, the *GCF* and *prime* *factorization* of two or more integers. Review "Classification and Definitions" and "Factors and Multiples" decks to be certain you know basic terms, concepts and rules.

Answer 61

* If the problem and the answer choices contain variables, you can substitute them with small numbers * If you can't solve it with straightforward math, and the answers have numerical values, try working backwards from them

Answer 62

We need to find the number of unique prime factors in the square of *k* \* A. You can easily solve it algebraically. If you decide to pick numbers for variables, make sure they fit the problem. (*k \* A*)² = (*k \* m \* n*)² The product of k and A has three unique prime factors. Squaring the product will change the number of factors, however, it will not change the number of *unique* prime factors. The square of the product of k and A has *three unique prime* number factors.

Answer 63

Whether you solve it algebraically or pick numbers, beware of the answers that are too obvious. Let's find *X* by figuring out the product of three unique prime numbers greater than 2. *X* = 3 x 5 x 7 = 105. The obvious factors of 105 are 1, 3, 5, 7, and 105. So are there 5 factors? No. (a) choice is a trap. What about 15 (3 x 5), 21 (3 x 7), and 35 (5 x 7)? 105 is also divisible by 15, 21 and 35, i.e. the products of any two of the three factors. 105 has 8 positive factors, choice (d). Algebraic method: *X = k \* m \* n*, where *k, m*, and *n* are prime factors greater than 2. Factors of *X* are *k, m, n, k \* m, k \* n, m \* n, k \* m \* n.* And 1 - a factor of every number.

Answer 64

You can replace variables with numbers but remember you are looking for all possible ways to make the sum an *even* number. Let's test the triples below: (3, 4, 5) , (4, 5, 6) , (2, 4, 6) , (1, 3, 5) Only the combinations with either one even number or three even numbers result in an even sum. The correct answer is (i) and (iii). Another way to solve this problem is to express odd as even + 1. We need to test (i) and (ii) since (iii) obviously works. Even + Odd + Odd = Even + Even + 1 + Even + 1 = 3 Even + 2 is definitely even. Even + Even + Odd = Even + Even + Even + 1 is odd.

Answer 65

This problem is pretty straightforward if you know what you really need to find in order to find the time the traffic lights will all change again. You are looking for the Least Common Multiple (LCM) of 12, 36 and 60. Remember how to find the LCM by using prime factorization? * 12 = 2 x 2 x 3 * 36 = 2 x 2 x 3 x 3 * 60 = 2 x 2 x 3 x 5 * LCM = 2 x 2 x 3 x 3 x 5 = 180 180 seconds is 3 minutes. So, the next time all three lights change simultaneously is 9:03 a.m.

Answer 66

The backsolving strategy will help you find the right answer in seconds. Start with the middle choice (c) and work your way up or down. Choice (a) meets all the requirements of the problem. Logically, if *P* - 1 is divisible by both 4 and 6, this number is a multiple of 12. With that in mind, examine the answers to find multiples of 12 + 1. (a), (c), and (e) work. However, only (a) 13 fits the problem.

Answer 67

This problem lends itself to using the picking numbers strategy once you've analyzed the information thoroughly. A, B, C and D are different digits. Both DC and CD are two digit numbers; therefore, the hundredth digit in their sum ABA is 1. So, A = 1. Re-write the problem. D C _+ C D_ 1 B 1 The sum of C and D has to equal 11. Single digit numbers that add up to 11 are (2;9) (3;8) (4;7) (5;6). Remember, all the digits have to be different. So, B can not be 1 or be one of the digits already used for D or C. Try all the pairs. (2; 9) does not work. 2 is not part of the answer choices but 9 is. So, it's (e).

Answer 68

There is a variable in the answer choices so we can pick a small and admissible number to replace it. Let *n* equal 4 and plug it in to check every answer choice. Be careful with choice (c). When *n* = 4, the result is odd. When *n* = 8, it becomes even. Choice (d) is the correct answer. Logically, an even number multiplied by any number is even. This eliminates (a), (b) and (e). An odd number multiplied by an odd number is always odd. That's choice (d).

Answer 69

"Parts" of the ratio add up to the "whole". Depending on the problem at hand you might need to convert a part-to-part ratio into a part-to-whole ratio or vice verse. Well, if you don't know how numbers or variables relate to each other, you might use the wrong formula and get a wrong answer. *Example*: If you drive *twice* as fast, you will go *twice* as far in a given time. Speed and distance are *directly* proportional. If you drive *twice* as fast, you need *half* the time to go the distance. Speed and time are *inversely* proportional.

Answer 70

Determine whether: * ratios are *part-to-part* or *part-to-whole* * numbers (variables) relate *directly* or *inversely* * you know actual quantities of items that correspond to parts of the ratio * the total quantity of items Set up a proportion with two or more ratios from the problem and solve by cross-multiplying.

Answer 71

Analyze this problem according to the recommended strategy. The ratio of almonds to cashews is 7 to 4. The mixture weighs 16.5 pounds. The ratio is a part-to-part ratio but the actual quantity you are given is the total weight of the mixture. Therefore, you need to convert the ratio into part-to-whole. "Parts" add up to "whole". 7 + 4 = 11 so the ratio of cashews to total is 4 to 11. Let's set up a proportion. ⁴/₁₁ = *^x*/_16.5 ⇒ *x* = 6 The weight of cashews in the mixture is 6 lbs.

Answer 72

# * _State the purpose:_ Set up proportions, cross-multiply * _Define the question:_ Find the number of songs erased * _Extract key info:_ You had 91 pop songs The original ratio *R : P : R&B* = 6 : 13 : 4 The new ratio *R : P : R&B* = 4 : 17 : 7 * _Identify strategy_: Determine the type of ratio (part-to-part) and set up proportions to find the original and the new number of songs in each genre original *R : P : R&B* = 6 : 13 : 4 original *R : P : R&B* = *x* : 91 : *y* ⇒ 13 : 91 = 6 : *x* = 4 : *y* ⇒ *x* = 42, *y* = 28 The number of R&B songs remained the same. However, some songs were erased, changing the total and the numbers in the ratio. new *R : P : R&B* = 4 : 17 : 7 new *R : P : R&B* = *m : n* : 28 ⇒ 7 : 28 = 17 : *n* = 4 : *m* ⇒ *n* = 68, *m* = 16 Original total was 161. New total is 112. *You erased 49 songs*. The answer matches the question so you are done.

Answer 73

Use the cross-multiplication method to re-write this proportion. * a : b = c : d ⇒ ad = bc* (1) , (2), and (3) answer choices result in the same equal products. The only answer choice that does *not* yield the original statement is choice (4). *dc = ba* is not the same as *ad = bc.*

Answer 74

* _The purpose:_ Set up an inverse proportion * _The question:_ Which expression can be used to find the weight of an astronaut? * _Key info:_ the weight varies *inversly* as the *square* of the distance * _Formulas and strategies:_ Know the definition and the formula for the inverse proportion. *y = ^k/_x* * _Logical conclusion:_ *d²* has to be in the denominator of the correct expression. Choice (b) is the correct answer.

Answer 75

Analyze the problem according to the recommended method and strategy. M = 10, W = 14 in one class and therefore, the ratio of M to W in the first class is 5 to 7. The ratio is a part-to-part ratio. "Parts" add up to the "whole". 5 + 7 = 12. Since the ratios of dancers in two classes are equal, the number of people in the second class has to be a multiple of 12. Choice (a) is correct.

Answer 76

A good understanding of the concept of the probability of an event happening is a must. You need to know the difference between: * *simple* (probability of one event) and *compound* (probability of two or more events) * *dependent* and *independent* events * *simultaneous and mutually exclusive* events If these concepts are unfamiliar to you, review the mini-tutorials in the "Fractions and Probability" deck.

Answer 77

The probability of randomly choosing a red bead from a bowl of beads is ³*/*₇. According to the definition of the probability, it means that there is a total of 7 beads in the bowl, of which 3 are red. Thus, the numerator can be any multiple of 3 and the denominator any multiple of 7. Evaluate the answer choices to find a number that is *not* a multiple of 7. It's choice (c) 81.

Answer 78

Use the recommended approach to problem solving to analyze the information. Do it on your own this time. The probability of picking an odd number is ¹³*/*₁₉ which means that there are 13 odd numbers out of a total of 19 numbers on the list. It also means that there are 6 even numbers in the total. So, ⁶*/*₁₃ would be the ratio of even to odd numbers in this list.

Answer 79

The washing phase takes 50% of total time, rinsing takes 30% and drying takes the rest of the time (20%). You should know that the probability of an event(s) *not* happening is **P not A = 100% - P (A)** 100% - 50% - 20% = 30% The probability that the dishwasher is not in a washing or drying phase is 30%.

Answer 80

This is a typical word problem...long and confusing. To help you digest the information, we've broken it into 4 blocks and underlined key units. Test makers will not do that for you! *Active reading* is an important technique (reviewed in this deck) that keeps you focused. Analyze key facts to make the following conclusions: Total *= S + B + W* *S = 3B + 3W ⇒ T = 4B + 4W* P(*W*) = 3 P(*B*). It means that the number of white socks is 3 times the number of black socks. If *W* = 6, *B* = 2. Total = 4 \* 2 + 4 \* 6 = 32

Answer 81

* _The purpose:_ Evaluate the probability of 3 events * _The question:_ Find the probability that *all* three students like pizza * _Key info:_ 4 out of 6 students like pizza. 3 students are chosen at random *with "replacement"* * _Logic conclusion_: The probability that *any one* of three students chosen likes pizza is ⁴*/*₆ = ²*/*₃. The events are independent. The total number of possible outcomes is the same for each choice * _Formula and strategy:_ Know the meaning of the words *with "replacement*". Know to *multiply* individual probabilities to find the probability that *all* three students like pizza ²/₃ x ²/₃ x ²/₃ = ⁸/₂₇ - choice (e) * _Is it the answer to the question?_ Yes.

Answer 82

* _The purpose:_ To evaluate the probability of two events * _The question:_ Find the probability of pulling a black t-shirt and then a white t-shirt * _Key info:_ There are 3 multicolored, 5 black and 6 white t-shirts. You pull shirts *without replacement* * _Formulas and strategy:_ Know the definition of *dependent* events. Know to multiply individual probabilities if both events must happen * _Logical conclusions:_ There are a total of 14 t-shirts in the closet. The probability of pulling a black t-shirt is ⁵*/*₁₄. The probability of then pulling a white t-shirt without replacement is ⁶*/*₁₃ * _Apply all knowledge_: *Multiply* the two to find the probability of both events ⁵*/*₁₄ x ⁶*/*₁₃= ¹⁵*/*₉₁

Answer 83

You need to follow the recommended process. However, challenging, long word problems require you to do a few *extra* things to process the information easier. Improve your *active reading* skills to make sure your brain registers every important detail! *Reminder*: Active reading involves reading the problem aloud and re-telling it. It also involves highlighting or underlining key information as well as creating a simple *diagram* or mapping out the information.

Answer 84

SAT test makers love *distance*, *age*, *percentage*, and *work* word problems. ## Footnote Identifying the type of problem will help trigger your brain to think of what formula or what approach to use to solve it.

Answer 85

When solving long, challenging word problems, you should *not*: * Read the first sentence and start solving the problem * Disregard our advice on active reading * Write an equation without thinking the problem through

Answer 86

That's an easy problem. Some of you might prefer to work with small, simple numbers instead of variables. *n* = 10, *y* = 2, *z* = 3 Re-read the problem using numbers instead of variables. There are 10 socks in 2 drawers of Alex's closet. How many socks can there be in 3 drawers? (10 ÷ 2) \* 3 = 15. Now, plug the variables back in. *(n ÷ y )* \* *z = ^nz/_y*

Answer 87

Follow the steps of the critical thinking process. Employ the active reading strategy. Pick small numbers to replace the variables if it helps. *n* = 18, *y* = 3,* c = 2* The problem makes more sense once you re-read it with numbers instead of variables. (18 ÷ 3) - 2 = 4. Write the expression to figure out what percent of 18 is 4.... ⁴/₁₈ x 100% Now, plug the variables back in... ^{(n ÷ y)- c}*/_n*\* 100% ⇒ *^{100(n - cy)}/_ny* This expression shows what percent of socks are left in each drawer out of the total number of socks.

Answer 88

Highlight key units of information and map out the problem. The algebraic solution is probably easier in this case but if you are more comfortable with numerals, replace the variables with manageable numbers. * *^x/₅* - original $$ each is supposed to pay * *^x/_{5 + y}* - $$ each has to pay after more people decided to contribute * *^x/₅ - ^x/_{5 + y}* - the $$ difference Don't get discouraged if the expression you came up with is not among the answer choices. Think how to re-work the fractions above. Find the LCD of the two and simplify to get to the correct answer (e) *^xy/_{25 + 5y}*.

Answer 89

Use active reading tips to really read the problem. Re-tell and underline or highlight key facts. Create a simple diagram to better sort out the information as shown below. Core knowledge you need here is the distance formula. Make sure the rates have the same unit measures as the other components. **Distance = Speed x Time** * Distance (D₁) from NYC to the rest area is 150 miles (2.5 x 60) * The speed on the 2nd part of the trip is 66 mph. * It takes the driver 3 hours (6 - 2.5 - 0.5) to cover the distance between the rest area and Boston. That distance (D₂) is 198 miles (3 x 66) The total distance beween NYC and Boston is the sum of D₁ and D₂ = 198 + 150 = 348

Answer 90

Employ the recommended step-by-step method as well as the active reading strategy. Practice both on your own. Core knowledge: You have to know the *average speed* formula and realize what the average speed is *not*. The average speed is *NOT* the average of speeds on different parts of the trip. **Avg Speed = ^{Total Distance}*/*_{Total Time}** Total distance is 804 miles (480 + 324). It took Cooper family 16 hours (8 + 6 + 2) to cover that distance. Divide total distance by total time to come up with the average speed of the entire trip. It equals 50.45 mph.

Answer 91

This problem involves drawing a *Venn diagram,* which is usually made up of two overlapping circles. It would be a mistake to just add 22, 21, 15 and 3. You would be counting some people twice. 15 kids play both sports; that's the intersection of two sets. of kids who like *only* soccer = 22 - 15 of kids who like *only* football = 21 - 15 The total # of kids = *only* soccer + *only* football + *both* + *neither* 7 + 6 + 15 + 3 = 31

Answer 92

Extract key facts from the problem with the help of active reading method. Draw a 3-circle *Venn diagram*. No student plays three instruments, so zero goes where the 3 sets overlap. 8 students play two instruments. It doesn't matter how you break it down as long as the areas where 2 sets intersect contain a total of 8 people. guitar *only* → 23 - 1 - 2 = 20 students piano *only* →15 - 1 - 1 = 13 students violin *only* → 20 - 1 - 2 = 17 students Total = 20 + 13 + 17= 50 Or.... you can do it this way. The total # of students in the school is comprised of the # of students who play *one* instrument *minus* the # of students who play *two* instruments. You subtract that # because you don't want to count them twice. Total = 23 + 20 + 15 - 8 = 50

Answer 93

The ages of all four siblings are interrelated but not in the obvious way. Let's map out the problem using math "language". Kate is older than Chris: *K \> C* Michael is younger than Kate: *M \< K ⇒ K \> M* Chris is older than you: *C \> Y* Based on the information given, we can conclude that K \> C \> you. Evaluate the answer choices. Obviously, if Kate is older than you, you *cannot* be older than Kate, choice (d).

Answer 94

* _The purpose:_ Add the individual work rates to find the combined work rate * _The question:_ Find the time it takes all 3 machines working together to complete the job * _Key info:_ The 1st copy machine takes 4 hrs to complete the job, the 2nd takes 3 hrs, the 3rd takes twice as long as the 2nd * _Formula and strategy:_ Use the active reading method and the knowledge of the combined work formula. ***^x/_{T together} = ^x/_Ta + ^x/_Tb + ^x/_Tc*** * X* is the total job*. T_a,b,c* is the *time* that *each* machine by itself requires to complete the total job. T _together is the *time* *all* 3 machines require to complete the job working together * _Apply all knowledge:__ _Since you don't know what the job is, assume it's 100% or 1. ¹/_{T together} = ¹*/*₄ + ¹*/*₃ + ¹*/*₆ = ⁹*/*₁₂ T _together = ¹²*/*₉ = 1 ¹/₃ * _Have you answered the question? Does your answer make sense?_ Yes

Answer 95

Use the active reading method to extract key information and diagram or map out the problem. Use your knowledge of core concepts such as the formulas for the average speed and the distance. **Speed x Time = Distance** **Avg Speed = ^{Total Distance}*/*_{Total Time}** Since the unit rate of speed is in miles per hour and the time is given in minutes, convert the time into hours. 1st person: 6 x 1.5 = 9 miles 2nd person: 9 x ²*/*₃ = 6 miles Each lap is 1/4 of a mile, therefore, two friends ran a *total of 60 laps* (9 + 6) x 4. Together, both friends ran 15 miles in 2.17 hours. *Avg speed* = ¹⁵*/*_2.17≈ *6.9 mph*

Answer 96

On your own, analyze the information using the step-by-step approach. Read actively and diagram (visual helps sort out the information). *A* = Alex, *G* = grandfather. * (*A* - 1) \* 8 = *G* + 4 * *A + G* = 87 Express *G* in terms of *A* in the 1st equation and plug into the 2nd equation. 8A - 8 = *G* + 4 ⇒ *G* = 8*A* - 12 *A* + 8*A* - 12 = 87 9*A* = 99 ⇒ *A* = 11 ⇒ *G* = 76 *G - A* = 76 - 11 = 65. Alex is 65 years younger than his grandfather.

Answer 97

How many possible ways can more than one event occur? Use the *Basic Counting Principle* to determine it. of possible ways = *p \* (p + 2) \* (p - 2)* = *p*(*p²* - 4) = *p³* - 4*p* You can order your dinner *p³* - 4*p* different ways.

Answer 98

* _The purpose:_ Figure out and add the discounted prices of *n* pairs of jeans * _The question:_ Find the formula that expresses the total cost of *n* pairs * _Key info:_ The 1st pair gets a 20% discount off the original price *x*. On each additional pair you receive an extra 10% discount. Quantity *n* \> 1. * _Conclusions:_ The price of the 1st pair is 80% of *x* or 0.8*x*. The price of each additional pair is 70% of *x* or 0.7*x*. The trick here is to realize that at 30% off you buy (*n* - 1) pairs. * _Formulas and strategies:_ Solve algebraically or pick numbers. The cost of n pairs is 0.8*x* + 0.7*x* (*n* - 1). Distribute the expressions to get the final answer 0.1*x* + 0.7*xn*, choice (d). Or pick $100 for *x* and 4 for *n*, re-read the problem and solve. $80 + $70 \* 3 = $290. Plug the numbers into the answer choices. The correct expression will give you the same result.

Advanced SAT Type Questions Flashcards

This deck tests your core knowledge of all subjects reviewed in other decks with the advanced SAT type questions. The deck provides a recommended approach as well as key strategies for successful problem solving.