Fractions & Decimals

Question 1

Fractional Change

Answer 1

“What is the fractional change in the price of this item?”

→ This is asking for the percentage change but just presented in fraction
→ e.g., $250 vs. $400 = 60% increase or 3/5 fractional change

Question 2

If a fraction is between 0 and 1, adding a positive constant to the numerator and denominator will make the fraction __________.

Answer 2

Larger

(and vice versa as long as the new numerator and denominator are still both positive)

Question 3

If a fraction is > 1, adding a positive constant to the numerator and denominator will make the fraction _________.

Answer 3

Smaller

(and vice versa as long as the new numerator and denominator are still both positive)

Question 4

Terminating Decimal Rule

Answer 4

The decimal equivalent of a fraction will terminate if and only if the denominator of the reduced fraction has a prime factorization that contains only 2s or 5s, or both.

Question 5

Rule for fraction that either terminates or repeats

Answer 5

Any fraction, in lowest terms, with an integer numerator and non-zero integer denominator will either terminate or repeat.

e.g., √⅓ = 1 / √3 → The denominator is not an integer. Thus, √⅓ neither terminates nor repeats.

Every rational number a/b has a terminating or repeating decimal representation

Question 6

Proper vs. Improper Fraction

Answer 6

• Proper fraction: numerator < denominator
• Improper fraction: numerator > denominator

Question 7

Bow Tie Method

Answer 7

Given two positive fractions a/b and c/d,

a/b > c/d if ad > bc

Question 8

Common Numerator Comparison Method

Answer 8

If all the fractions share a common numerator,

the larger the denominator, the smaller the fraction

Question 9

Division Formula

Answer 9

x = Qy + r

x / y = Q + r / y

• x = dividend
• y = divisor
• Q = integer quotient
• r = nonnegative remainder

Question 10

Types of Decimals

Answer 10

• Terminating decimals: 0.12, 3.4
• Repeating decimals: 3.444…., 0.121212….
• Non-terminating and non-repeating decimals: 0.12345…, 0.1010010000001….

Question 11

Leading Zero

Answer 11

Any zeros that appear after the decimal point and before the first nonzero digit.

• e.g., 0.731 has no leading zeros
• e.g., 0.002 has 2 leading zeros

Question 12

Leading Zero: Rules

Answer 12

If x is an integer with k digits, and if X is not a perfect power of 10, then 1/x has k-1 leading zeros

• e.g., 1/2 = 0.5 → no leading zeros
• e.g., 1/20 = 0.05 → 1 leading zero

If x is an integer with k digits, and if X is a perfect power of 10, then 1/x has k-2 leading zeros

• e.g., 1/10 = 0.1 → no leading zeros
• e.g., 1/100 = 0.01 → 1 leading zero

Question 13

1/6 in decimal

Answer 13

0.1666….

Question 14

2/3 in decimal

Answer 14

0.666…..

Question 15

5/6 in decimal

Answer 15

0.833…..

Question 16

1/7 in decimal

Answer 16

0.143 (non-terminating)

Question 17

2/7 in decimal

Answer 17

0.286 (non-terminating)

Question 18

3/7 in decimal

Answer 18

0.429 (non-terminating)

Question 19

4/7 in decimal

Answer 19

0.571 (non-terminating)

Question 20

5/7 in decimal

Answer 20

0.714 (non-terminating)

Question 21

6/7 in decimal

Answer 21

0.857 (non-terminating)

Question 22

1/9 in decimal

Answer 22

0.111 (non-terminating)