Fractions and Ratios Flashcards
How to compare fractions with the same denominator or numerator?
e.g.: which fraction is greater: 3/4, 2/4, 1/4? which fraction is greater: 3/6, 3/7, 3/8?
The greater the numerator, the greater the fraction 3/4 > 2/4 > 1/4
The greater the denominator, the smaller the fraction 3/6 > 3/7 > 3/8
If the numerator and the denominator are the same, what is the result of the fraction?
1
What to do if the numerator is greater than the denominator? (improper fraction)
You can rewrite the fraction as a sum of the integer part and the fractional part:
e.g. 5/4 = 1 + 1/4
What is a proper fraction?
Is when the numerator is smaller than the denominator (e.g., 3/4). The result will be a value between 0 and 1.
What is a ratio?
It is a part-to-part relationship that can be written in the form of a fraction. E.g.:
For every 2 bananas in a certain basket of fruit, there are 3 apples
= 2/3
This is a part to part relationship because 3 is not the whole. Please note that for each basket you will actually have 5 parts forming the whole.
What to do when adding fractions with the same denominator?
Add the numerators and simply repeat the denominator.
The same applies to subtraction.
(3a / b) + (4a / b) = 3a + 4a / b = 7a / b
What to do when adding fractions with different denominators?
Find a common denominator by renaming one of the fractions.
To rename a fraction you’ll need to multiply the top and bottom by the same number
1 / 4 + 3 / 8 = (1x2 / 4x2) + 3 / 8 = 2 / 8 + 3 / 8 = 5 / 8
A different way of doing that is by applying the double-cross method
How to use the double-cross method to compare fractions?
Multiply “up” the arrows, but ignore the denominator. Put the resulting number at the top of each respective arrow. Then compare the numbers. The side with the bigger number is the bigger fraction.
3 / 4 or 4 / 7?
3 x 7 = 21 vs. 5 x 4 = 29
21>20 so 3 / 5 is grater than 4 / 7
How to transform from a improper fraction to mixed number?
Break up the numerator to get an integer and a “left-over” fraction
13 / 4 =
3 ¹/4
How to revert and rewrite a mixed number as an improper fraction?
5 ²/3 = 5 + 2 / 3 (first rewrite the mixed number as a sum)
= 5 / 1 + 2 / 3 (rewrite the integer as a fraction with denominator 1)
= (15+2) / 3 = 17 / 3 (use the double-cross method to get into the common denominator)
Shortcut (take the digits of the mixed number):
Fraction numerator = (denominator) x (integer) + (numerator)
Fraction denominator = the same of the fraction in the mixed number
eg: (3 x 5 + 2) / 3 = 17 / 3
How to simplify a fraction?
Find the common factors from the numerator and denominator and cancel out them
6 / 9 =
(2 x 3) / (3 x 3) =
2 / 3
(since 3 is a common factor on the to and bottom, cancel it)
What is the relation between ratio and simplification of a fraction?
A ratio is similar to a fraction in its simplest form:
If to each 2 apples you have 3 bananas, this means that
6 / 9 = 4 / 6 = 2 / 3
How to calculate ratio vs. actual numbers?
You will need to find the unknown multiplier, that will be always the same for both parts and also for the whole:
2 bananas for each 3 apples
Bananas / Apples = 2 / 3
Bananas / Apples = 2y / 3y
If you know the actual number for one of the parts (apple or bananas) you’ll be able to find the unknown multiplier and then get the actual figures in the ratio:
If I have 15 apples
3y = 15
y = 15 / 3
7 = 5
So the number of bananas will be:
2 * y = 2 * 5 = 10
How to multiply fractions?
To multiply fractions, multiply the tops together and multiply the bottoms together:
1 / 2 x 3 / 4 = 1 x 3 / 2 x 4 = 3 / 8
However, if possible, cancel factors (simplify the fraction) before you multiply. You are always allowed to cancel across the fractions as long as the two fractions are multiplied or divided (do not cancel across the fractions, however, if the two fractions are added or subtracted).
How to quickly multiply any number by 5?
5 is a fraction of 10/2, so you can multiply the number by 10 and then divide it by 2:
42 x 5 = 42 x 10 / 2 = 420 / 2 = 210
