SSAT Middle Level Quiz: Equivalent Fractions

20 questions · exam conditions

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Equivalent FractionsQuestion 1 of 20

Select the fraction equivalent to $\dfrac{5}{8}$ .

\dfrac{10}{16}

\dfrac{5}{16}

\dfrac{8}{5}

\dfrac{9}{16}

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SSAT Middle Level Quiz

SSAT Middle Level Quiz: Equivalent Fractions

Practice Equivalent Fractions in SSAT Middle Level with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Equivalent Fractions, giving you a quick way to practice the rules, question types, and explanations that matter most for SSAT Middle Level.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Select the fraction equivalent to $\dfrac{5}{8}$ .

$\dfrac{10}{16}$ (correct answer)
$\dfrac{5}{16}$
$\dfrac{8}{5}$
$\dfrac{9}{16}$

Explanation: This question tests SSAT Middle Level math skills, specifically finding equivalent fractions. Equivalent fractions are different fractions that represent the same value or proportion. They are found by multiplying or dividing both the numerator and denominator by the same number. In this problem, students select the fraction equivalent to 5/8, such as 10/16. The correct answer is valid because multiplying both the numerator and denominator of 5/8 by 2 gives 10/16. A common mistake, as seen in choice C, is to invert to 8/5. To help students, encourage simplifying to check. Practice with pie models.

Question 2

How can you express $\dfrac{3}{5}$ with denominator $15$ ?

$\dfrac{6}{15}$
$\dfrac{9}{15}$ (correct answer)
$\dfrac{3}{15}$
$\dfrac{15}{3}$

Explanation: This question tests SSAT Middle Level math skills, specifically finding equivalent fractions. Equivalent fractions are different fractions that represent the same value or proportion. They are found by multiplying or dividing both the numerator and denominator by the same number. In this problem, students express 3/5 with denominator 15, resulting in 9/15. The correct answer is valid because multiplying both the numerator and denominator of 3/5 by 3 gives 9/15. A common mistake, as seen in choice A, is to use the wrong multiplier, getting 6/15 which is 2/5. To help students, show how to divide denominators to find the factor. Encourage practice with fraction strips.

Question 3

A recipe calls for $\frac{2}{3}$ cup of flour. If James wants to make $\frac{3}{4}$ of the recipe, which fraction represents the amount of flour he needs, expressed in an equivalent form with denominator 24?

$\frac{8}{24}$
$\frac{12}{24}$ (correct answer)
$\frac{16}{24}$
$\frac{18}{24}$
$\frac{20}{24}$

Explanation: This problem combines fraction multiplication with finding equivalent fractions, two foundational skills you'll see throughout middle-level math. To find how much flour James needs, you multiply the original amount by the fraction of the recipe he's making:

\frac{2}{3} \times \frac{3}{4}

. When multiplying fractions, multiply the numerators together and the denominators together:

\frac{2 \times 3}{3 \times 4} = \frac{6}{12}

. Now you need to express

\frac{6}{12}

with denominator 24. Since

12 \times 2 = 24

, multiply both the numerator and denominator by 2:

\frac{6 \times 2}{12 \times 2} = \frac{12}{24}

. This confirms answer choice B is correct. Let's examine why the other choices are wrong. Choice A (

\frac{8}{24}

) might result from incorrectly calculating

\frac{2}{3} \times \frac{3}{4}

\frac{2 \times 4}{3 \times 3} = \frac{8}{9}

, then converting incorrectly. Choice C (

\frac{16}{24}

) could come from adding instead of multiplying:

\frac{2}{3} + \frac{3}{4} = \frac{8}{12} + \frac{9}{12} = \frac{17}{12}

, then making conversion errors. Choice D (

\frac{18}{24}

) might result from converting

\frac{3}{4}

to twenty-fourths (getting

\frac{18}{24}

) but forgetting to multiply by

\frac{2}{3}

. Remember: when you see "part of a recipe" problems, you're almost always multiplying fractions. Always simplify your result first, then convert to the requested denominator—this makes the arithmetic easier and reduces errors.

Question 4

If $\frac{a}{b} = \frac{12}{16}$ and $a$ and $b$ have no common factors other than 1, what is the value of $a + b$ ?

7 (correct answer)
12
16
28
35

Explanation: This question tests your understanding of equivalent fractions and reducing fractions to lowest terms. When a fraction is in lowest terms, the numerator and denominator share no common factors other than 1. You're given that

\frac{a}{b} = \frac{12}{16}

, but you need to find the values of

a

and

b

when they have no common factors other than 1. This means you need to reduce

\frac{12}{16}

to its simplest form. To reduce

\frac{12}{16}

, find the greatest common factor (GCF) of 12 and 16. The factors of 12 are 1, 2, 3, 4, 6, 12, and the factors of 16 are 1, 2, 4, 8, 16. The GCF is 4. Dividing both numerator and denominator by 4:

\frac{12÷4}{16÷4} = \frac{3}{4}

. Therefore,

a = 3

and

b = 4

, giving us

a + b = 3 + 4 = 7

. Looking at the wrong answers: (B) 12 is just the original numerator, ignoring the need to reduce the fraction. (C) 16 is just the original denominator, making the same error. (D) 28 comes from adding the unreduced values:

12 + 16 = 28

, which fails to recognize that the fraction must be in lowest terms. Strategy tip: When you see "no common factors other than 1," immediately think "lowest terms." Always reduce fractions completely by finding the GCF of the numerator and denominator, then divide both by that GCF.

Question 5

Which pair of fractions represents the same value when both are written in lowest terms?

$\frac{14}{21}$ and $\frac{18}{24}$
$\frac{15}{25}$ and $\frac{20}{35}$
$\frac{16}{20}$ and $\frac{24}{30}$ (correct answer)
$\frac{12}{18}$ and $\frac{20}{32}$
$\frac{9}{15}$ and $\frac{22}{33}$

Explanation: When you encounter fraction comparison problems, you need to reduce each fraction to its simplest form by dividing both the numerator and denominator by their greatest common factor (GCF). Let's work through option C first: For

\frac{16}{20}

, both 16 and 20 are divisible by 4, giving us

\frac{4}{5}

. For

\frac{24}{30}

, both 24 and 30 are divisible by 6, which also gives us

\frac{4}{5}

. Since both fractions reduce to the same value, C is correct. Now let's check why the other options don't work. In option A,

\frac{14}{21}

reduces to

\frac{2}{3}

(dividing by 7), while

\frac{18}{24}

reduces to

\frac{3}{4}

(dividing by 6). These are different values. For option B,

\frac{15}{25}

becomes

\frac{3}{5}

(dividing by 5), but

\frac{20}{35}

becomes

\frac{4}{7}

(dividing by 5). Again, different values. In option D,

\frac{12}{18}

simplifies to

\frac{2}{3}

(dividing by 6), while

\frac{20}{32}

simplifies to

\frac{5}{8}

(dividing by 4). These don't match either. Remember to always find the GCF systematically—list the factors of both numbers or use the division method. A quick check is to cross-multiply the reduced fractions: if they're equal, the cross products will be the same.

Question 6

Tom simplifies $\frac{36}{48}$ by dividing both the numerator and denominator by 6, getting $\frac{6}{8}$ . What should he do next to express this fraction in lowest terms?

Divide both numerator and denominator by 2 to get $\frac{3}{4}$ (correct answer)
Divide both numerator and denominator by 3 to get $\frac{2}{5}$
Multiply both numerator and denominator by 2 to get $\frac{12}{16}$
Divide the numerator by 6 and denominator by 8 to get $\frac{1}{1}$
The fraction $\frac{6}{8}$ is already in lowest terms and needs no change

Explanation: When you need to simplify fractions to lowest terms, you're looking for the greatest common factor (GCF) of the numerator and denominator, then dividing both by that number. Sometimes this happens in steps, as Tom discovered. Tom correctly started by dividing both 36 and 48 by 6 to get

\frac{6}{8}

. Now he needs to check if

\frac{6}{8}

can be simplified further. To do this, find the GCF of 6 and 8. The factors of 6 are 1, 2, 3, 6, and the factors of 8 are 1, 2, 4, 8. The greatest common factor is 2. Dividing both numerator and denominator by 2:

\frac{6÷2}{8÷2} = \frac{3}{4}

. Since 3 and 4 share no common factors other than 1, this fraction is now in lowest terms. Choice A is correct because it properly identifies the next step in the simplification process. Choice B incorrectly suggests dividing by 3. While 6 is divisible by 3, 8 is not, so

\frac{6÷3}{8÷3} = \frac{2}{2.67}

doesn't work and certainly doesn't equal

\frac{2}{5}

. Choice C moves in the wrong direction by multiplying, which makes fractions larger rather than simpler. Choice D uses completely incorrect logic, dividing the numerator by one number and the denominator by a different number, which fundamentally changes the fraction's value. Study tip: Always check if your simplified fraction can be reduced further by finding common factors. A fraction is in lowest terms when the numerator and denominator share no common factors except 1.

Question 7

If $\frac{x}{21} = \frac{4}{7}$ , what is the value of $x$ ?

3
12 (correct answer)
16
28
84

Explanation: This is a proportion problem where you need to find the value of

x

that makes the two fractions equal. When you see an equation with two equal fractions, you can solve it using cross multiplication. To solve

\frac{x}{21} = \frac{4}{7}

, cross multiply by multiplying the numerator of each fraction by the denominator of the other fraction:

x \times 7 = 4 \times 21

. This gives you

7x = 84

. Dividing both sides by 7, you get

x = 12

. You can verify this by substituting:

\frac{12}{21} = \frac{4}{7}

. Since

\frac{12}{21}

simplifies to

\frac{4}{7}

(dividing both numerator and denominator by 3), the equation is true. Looking at the wrong answers: Choice (A) 3 would give you

\frac{3}{21} = \frac{1}{7}

, which doesn't equal

\frac{4}{7}

. Choice (C) 16 would give you

\frac{16}{21}

, which cannot be simplified to equal

\frac{4}{7}

. Choice (D) 28 would give you

\frac{28}{21} = \frac{4}{3}

, which is greater than 1 while

\frac{4}{7}

is less than 1. The correct answer is (B) 12. Study tip: When solving proportions, always cross multiply to clear the fractions, then solve the resulting linear equation. Double-check your answer by substituting back into the original equation to make sure both sides are equal.

Question 8

Marcus writes several equivalent fractions for $\frac{6}{9}$ : $\frac{12}{18}$ , $\frac{18}{27}$ , $\frac{24}{36}$ , and $\frac{30}{45}$ . Which pattern describes how he generated these fractions?

Multiply both numerator and denominator by consecutive integers starting with 2
Multiply both numerator and denominator by multiples of 6 in order
Add 6 to both numerator and denominator repeatedly
Multiply both numerator and denominator by 2, 3, 4, 5 respectively (correct answer)
Multiply the numerator by even numbers and denominator by odd numbers

Explanation: When you encounter questions about equivalent fractions and patterns, you need to analyze how the numerator and denominator change from the original fraction to each new fraction. Let's examine how Marcus generated each fraction from the original

\frac{6}{9}

. For

\frac{12}{18}

: the numerator went from 6 to 12 (multiply by 2) and denominator from 9 to 18 (also multiply by 2). For

\frac{18}{27}

: 6 becomes 18 (multiply by 3) and 9 becomes 27 (multiply by 3). Continuing this pattern:

\frac{24}{36}

uses a multiplier of 4, and

\frac{30}{45}

uses a multiplier of 5. The sequence is multiply by 2, then 3, then 4, then 5. Choice D correctly identifies this pattern: multiply both numerator and denominator by 2, 3, 4, 5 respectively. Choice A is wrong because the multipliers aren't just consecutive integers starting with 2 — they're consecutive integers, but the pattern starts with 2 and continues through 5. Choice B is incorrect because the multipliers (2, 3, 4, 5) aren't multiples of 6. Choice C is wrong because Marcus isn't adding 6 repeatedly; if he were,

\frac{6}{9}

would become

\frac{12}{15}

, not

\frac{12}{18}

. Remember that equivalent fractions are created by multiplying both parts by the same number. When analyzing fraction patterns, always check what number was used to multiply both the numerator and denominator for each step — this reveals the underlying pattern more clearly than looking at the fractions themselves.

Question 9

A fraction $\frac{p}{q}$ in lowest terms is equivalent to $\frac{48}{72}$ . If $p + q = 10$ , what is the value of $p$ ?

2
4 (correct answer)
6
8
12

Explanation: This question tests your understanding of equivalent fractions and how to reduce fractions to lowest terms. When you see a fraction that needs to be simplified, always look for the greatest common factor (GCF) of the numerator and denominator. To find the fraction

\frac{p}{q}

in lowest terms that's equivalent to

\frac{48}{72}

, you need to reduce

\frac{48}{72}

by finding the GCF of 48 and 72. The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. The GCF is 24. Dividing both numerator and denominator by 24:

\frac{48}{72} = \frac{48÷24}{72÷24} = \frac{2}{3}

p = 2

and

q = 3

. Let's verify:

p + q = 2 + 3 = 5

. Wait, this doesn't equal 10 as required! Looking more carefully, we need

p + q = 10

, so we need an equivalent fraction to

\frac{2}{3}

where the numerator and denominator sum to 10. If

\frac{p}{q} = \frac{2}{3}

, then

p = 2k

and

q = 3k

for some integer

k

. Since

p + q = 10

, we have

2k + 3k = 10

, so

5k = 10

, giving us

k = 2

. Therefore

p = 4

and

q = 6

. Choice A (2) gives you the reduced numerator but ignores the constraint that

p + q = 10

. Choice C (6) is the denominator of our final fraction. Choice D (8) doesn't correspond to any meaningful part of this problem. Strategy tip: When working with equivalent fractions, always check that your answer satisfies all given conditions, not just the fraction equivalence.

Question 10

Jenny needs $\frac{3}{8}$ cup of oil for a recipe. Her measuring cup shows equivalent fractions. Which of these equivalent fractions would also measure exactly $\frac{3}{8}$ cup?

$\frac{6}{16}$ cup (correct answer)
$\frac{9}{32}$ cup
$\frac{12}{24}$ cup
$\frac{12}{40}$ cup
$\frac{18}{64}$ cup

Explanation: When you encounter equivalent fractions, you're looking for fractions that represent the same value even though they have different numerators and denominators. To find equivalent fractions, you multiply or divide both the numerator and denominator by the same number. Starting with

\frac{3}{8}

, let's check each option systematically. For choice A,

\frac{6}{16}

, notice that both 6 and 16 are double the original values:

\frac{3 \times 2}{8 \times 2} = \frac{6}{16}

. This gives us the same value as

\frac{3}{8}

, so A is correct. Let's verify why the other choices don't work. Choice B,

\frac{9}{32}

, might seem promising since 9 is 3 × 3, but 32 isn't 8 × 3 (that would be 24). Instead, 32 = 8 × 4, so this fraction doesn't maintain the same ratio. Choice C,

\frac{12}{24}

, has 12 = 3 × 4, but 24 = 8 × 3, not 8 × 4. The numerator and denominator were multiplied by different numbers, breaking equivalence. Choice D,

\frac{12}{40}

, suffers the same problem: while 12 = 3 × 4, we have 40 = 8 × 5, so again different multipliers were used. Study tip: To quickly verify equivalent fractions, cross-multiply. For equivalent fractions

\frac{a}{b} = \frac{c}{d}

, you should get

a \times d = b \times c

. With

\frac{3}{8}

and

\frac{6}{16}

3 \times 16 = 48

and

8 \times 6 = 48

✓. This cross-multiplication check catches errors faster than trying to find the common multiplier.

Question 11

Which statement about equivalent fractions is always true?

Equivalent fractions have the same numerator when written in lowest terms
Equivalent fractions have the same denominator when written in lowest terms
Equivalent fractions have numerators and denominators that differ by the same amount
Equivalent fractions represent the same portion of a whole when compared visually (correct answer)
Equivalent fractions must have denominators that are multiples of each other

Explanation: When you encounter questions about equivalent fractions, focus on the fundamental definition: equivalent fractions represent the same value or portion, even though they may look different when written out. The correct answer is D because equivalent fractions, by definition, represent identical portions of a whole. Whether you're looking at

\frac{1}{2}

\frac{2}{4}

, or

\frac{3}{6}

, each represents exactly half of something. This visual equality is what makes fractions equivalent in the first place. Let's examine why the other choices fail. Choice A is incorrect because equivalent fractions rarely have the same numerator in lowest terms. For example,

\frac{2}{4}

and

\frac{3}{6}

are both equivalent to

\frac{1}{2}

, but their numerators (2 and 3) differ from the lowest-terms numerator (1). Choice B makes the same error with denominators—

\frac{4}{8}

and

\frac{6}{12}

are both equivalent to

\frac{1}{2}

, but their denominators (8 and 12) don't match the lowest-terms denominator (2). Choice C suggests that the difference between numerator and denominator stays constant, but this isn't true either. In

\frac{1}{2}

, the difference is 1, while in the equivalent fraction

\frac{3}{6}

, the difference is 3. Remember this key principle: equivalent fractions are created by multiplying or dividing both the numerator and denominator by the same non-zero number. This preserves the value while changing the appearance, which is why visual representation remains constant even when the numbers change.

Question 12

Two equivalent fractions have numerators that differ by 15 and denominators that differ by 25. If the smaller fraction is $\frac{9}{15}$ , what is the larger fraction?

$\frac{24}{40}$ (correct answer)
$\frac{21}{35}$
$\frac{18}{30}$
$\frac{15}{25}$
$\frac{12}{20}$

Explanation: When you encounter equivalent fractions with specific differences between their parts, you're working with proportional relationships. The key insight is that equivalent fractions maintain the same ratio, so if one fraction is a multiple of another, both the numerator and denominator are multiplied by the same factor. Starting with the smaller fraction

\frac{9}{15}

, you need to find what factor creates the given differences. If the numerators differ by 15, then the larger numerator is

9 + 15 = 24

. If the denominators differ by 25, then the larger denominator is

15 + 25 = 40

. This gives you

\frac{24}{40}

. To verify these fractions are equivalent, simplify both:

\frac{9}{15} = \frac{3}{5}

and

\frac{24}{40} = \frac{3}{5}

. They're equal, confirming your answer. Looking at the wrong choices: Choice B gives

\frac{21}{35}

, where the differences would be 12 and 20, not 15 and 25. Choice C gives

\frac{18}{30}

, with differences of 9 and 15 - close, but not matching the problem's requirements. Choice D gives

\frac{15}{25}

, with differences of 6 and 10. The correct answer is A)

\frac{24}{40}

. Strategy tip: For equivalent fraction problems, always check that your fractions actually reduce to the same value. Also, read carefully - problems often give you the differences directly, so simple addition from the starting values usually leads you to the answer quickly.

Question 13

Sarah has $\frac{5}{8}$ of a pizza left. She wants to divide it equally among 3 people. Which fraction, when expressed with a denominator of 48, represents each person's share?

$\frac{8}{48}$
$\frac{10}{48}$ (correct answer)
$\frac{15}{48}$
$\frac{16}{48}$
$\frac{20}{48}$

Explanation: When you encounter a problem involving dividing fractions, you're essentially performing fraction division, which means multiplying by the reciprocal. To find each person's share, you need to divide

\frac{5}{8}

by 3. Remember that dividing by 3 is the same as multiplying by

\frac{1}{3}

\frac{5}{8} \div 3 = \frac{5}{8} \times \frac{1}{3} = \frac{5}{24}

Now you need to express

\frac{5}{24}

with a denominator of 48. To convert, determine what number you multiply 24 by to get 48:

24 \times 2 = 48

. Multiply both numerator and denominator by 2:

\frac{5}{24} = \frac{5 \times 2}{24 \times 2} = \frac{10}{48}

Choice A (

\frac{8}{48}

) represents the mistake of thinking each person gets

\frac{1}{8}

of the original whole pizza, ignoring that Sarah only has

\frac{5}{8}

remaining. Choice C (

\frac{15}{48}

) comes from incorrectly multiplying

\frac{5}{8} \times 3

instead of dividing by 3. Choice D (

\frac{16}{48}

) results from converting

\frac{5}{8}

\frac{20}{48}

correctly but then dividing incorrectly. The key strategy here is to remember that "dividing equally among" always means division, and when dividing fractions, multiply by the reciprocal. Always double-check your final conversion by ensuring you multiply both parts of the fraction by the same number.

Question 14

Lisa writes the fraction $\frac{42}{56}$ and claims it equals $\frac{6}{8}$ because "I divided the top and bottom by 7." Is her work correct, and what is the fraction in lowest terms?

Yes, her work is correct, and $\frac{6}{8}$ is in lowest terms
Yes, her work is correct, but $\frac{6}{8}$ reduces further to $\frac{3}{4}$ (correct answer)
No, she should have divided by 14 to get $\frac{3}{4}$ directly
No, dividing by 7 gives $\frac{35}{49}$ , which reduces to $\frac{5}{7}$
No, $\frac{42}{56}$ cannot be reduced because 42 and 56 share no factors

Explanation: When you encounter fraction problems asking whether simplification work is correct, you need to check two things: is the arithmetic right, and is the result in lowest terms? Let's verify Lisa's work step by step. She started with

\frac{42}{56}

and divided both numerator and denominator by 7:

$42 ÷ 7 = 6$ ✓
$56 ÷ 7 = 8$ ✓

So Lisa correctly obtained

\frac{6}{8}

. However, this fraction isn't in lowest terms yet. To find the lowest terms, we need the greatest common factor (GCF) of 6 and 8. Since

6 = 2 × 3

and

8 = 2 × 4

, their GCF is 2. Dividing both parts by 2:

\frac{6}{8} = \frac{6÷2}{8÷2} = \frac{3}{4}

. Now let's examine each answer choice. Choice A incorrectly claims

\frac{6}{8}

is already in lowest terms—it's not, since both 6 and 8 are divisible by 2. Choice B correctly identifies that Lisa's arithmetic is right but notes that

\frac{6}{8}

reduces further to

\frac{3}{4}

. Choice C suggests Lisa should have divided by 14 instead, but

42 ÷ 14 = 3

and

56 ÷ 14 = 4

, which would work but doesn't make her original work wrong. Choice D contains calculation errors: dividing 42 by 7 gives 6, not 35. Study tip: Always check if a simplified fraction can be reduced further by finding the GCF of the numerator and denominator. A fraction is in lowest terms only when their GCF is 1.

Question 15

A proportion states that $\frac{5}{8} = \frac{x}{24}$ . After solving for $x$ , the resulting fraction $\frac{x}{24}$ is reduced to lowest terms. What is this reduced fraction?

$\frac{5}{8}$ (correct answer)
$\frac{15}{24}$
$\frac{5}{24}$
$\frac{3}{8}$
$\frac{15}{8}$

Explanation: When you encounter a proportion problem, you're working with two equal ratios. The key insight here is that proportions maintain their equality even after you solve for the unknown variable. To solve

\frac{5}{8} = \frac{x}{24}

, cross-multiply:

5 \times 24 = 8 \times x

, which gives you

120 = 8x

. Dividing both sides by 8 yields

x = 15

. So the fraction becomes

\frac{15}{24}

. Now you need to reduce this to lowest terms by finding the greatest common factor (GCF) of 15 and 24. The factors of 15 are 1, 3, 5, 15, and the factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The GCF is 3. Dividing both numerator and denominator by 3:

\frac{15 ÷ 3}{24 ÷ 3} = \frac{5}{8}

. Let's examine the wrong answers: Answer B (

\frac{15}{24}

) is the unreduced form of your solution—a common trap for students who forget the "reduce to lowest terms" instruction. Answer C (

\frac{5}{24}

) results from incorrectly keeping the original numerator without solving the proportion. Answer D (

\frac{3}{8}

) might come from calculation errors or confusion about which numbers to divide. The correct answer is A (

\frac{5}{8}

), which makes perfect sense: when you reduce a fraction that's part of a true proportion, you should get back to the original ratio. Strategy tip: In proportion problems asking for reduced fractions, expect the answer to equal one of the original fractions in the proportion—this serves as a useful check for your work.

Question 16

A recipe calls for ingredients in the ratio $\frac{2}{5} : \frac{4}{10}$ . What can be concluded about these two amounts?

The first amount is twice the second amount
The second amount is twice the first amount
The amounts are equal (correct answer)
The first amount is $\frac{1}{5}$ more than the second amount
The ratio cannot be simplified without knowing actual quantities

Explanation: When you encounter ratios with fractions, the key is to simplify both fractions first and compare their actual values. Don't let the different denominators fool you into thinking the amounts must be different. Let's simplify each fraction in the ratio

\frac{2}{5} : \frac{4}{10}

. The first fraction

\frac{2}{5}

is already in lowest terms. For the second fraction,

\frac{4}{10} = \frac{4÷2}{10÷2} = \frac{2}{5}

. So our ratio is actually

\frac{2}{5} : \frac{2}{5}

, which means the amounts are equal. You can also verify this by converting to decimals:

\frac{2}{5} = 0.4

and

\frac{4}{10} = 0.4

. Same value, so the amounts are equal. This confirms that answer choice C is correct. Let's examine why the other choices are wrong. Choice A claims the first amount is twice the second, which would mean

\frac{2}{5} = 2 \times \frac{4}{10}

, but

2 \times \frac{4}{10} = \frac{8}{10} = \frac{4}{5}

, not

\frac{2}{5}

. Choice B suggests the opposite relationship, which is equally incorrect. Choice D states the first amount is

\frac{1}{5}

more than the second, but since they're equal, the difference is zero, not

\frac{1}{5}

. Remember: always simplify fractions before comparing them. Different-looking fractions can represent the same value, and the SSAT often tests whether you recognize equivalent fractions in various forms.

Question 17

A rectangular garden has length $\frac{24}{32}$ meters and width $\frac{15}{20}$ meters. If both dimensions are expressed as equivalent fractions with the same denominator, what is the smallest possible common denominator?

32
20
16
8
4 (correct answer)

Explanation: This question tests your understanding of equivalent fractions and finding the least common denominator (LCD). When you need to express fractions with the same denominator, you're looking for the smallest number that both original denominators divide into evenly. First, simplify both fractions to lowest terms.

\frac{24}{32} = \frac{3}{4}

(dividing by 8) and

\frac{15}{20} = \frac{3}{4}

(dividing by 5). Wait - both fractions are actually equivalent to

\frac{3}{4}

! Since both dimensions equal

\frac{3}{4}

, the smallest common denominator is simply 4. You can verify this:

\frac{3}{4}

and

\frac{3}{4}

already share the denominator 4, which is the smallest possible. Looking at the wrong answers: Choice (A) 32 is one of the original denominators, but using it would mean writing

\frac{24}{32}

and

\frac{12}{32}

- this works but isn't the smallest possible. Choice (B) 20 is the other original denominator, giving you

\frac{15}{20}

and

\frac{15}{20}

- again, this works but isn't minimal. Choice (C) 16 would give you

\frac{12}{16}

and

\frac{12}{16}

- still larger than necessary. Choice (D) 8 would work as

\frac{6}{8}

and

\frac{6}{8}

, but 4 is even smaller. The correct answer is E (which must be 4, though it's not shown in your options). Strategy tip: Always simplify fractions first before finding the LCD - you might discover the fractions are equivalent, making the problem much easier than it initially appears.

Question 18

Which of the following shows $\frac{7}{12}$ written as an equivalent fraction with denominator 36, and then reduced back to lowest terms?

$\frac{7}{12} \rightarrow \frac{21}{36} \rightarrow \frac{7}{12}$ (correct answer)
$\frac{7}{12} \rightarrow \frac{21}{36} \rightarrow \frac{21}{36}$
$\frac{7}{12} \rightarrow \frac{28}{36} \rightarrow \frac{7}{9}$
$\frac{7}{12} \rightarrow \frac{14}{36} \rightarrow \frac{7}{18}$
$\frac{7}{12} \rightarrow \frac{35}{36} \rightarrow \frac{35}{36}$

Explanation: This question tests your ability to create equivalent fractions and reduce them to lowest terms. When working with equivalent fractions, you multiply or divide both numerator and denominator by the same number to maintain the same value. To convert

\frac{7}{12}

to a denominator of 36, you need to determine what to multiply 12 by to get 36. Since

12 \times 3 = 36

, you multiply both the numerator and denominator by 3:

\frac{7}{12} = \frac{7 \times 3}{12 \times 3} = \frac{21}{36}

. To reduce

\frac{21}{36}

to lowest terms, find the greatest common factor (GCF) of 21 and 36. The factors of 21 are 1, 3, 7, 21, and the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. The GCF is 3. Dividing both numerator and denominator by 3:

\frac{21}{36} = \frac{21 ÷ 3}{36 ÷ 3} = \frac{7}{12}

. Choice A correctly shows this process:

\frac{7}{12} \rightarrow \frac{21}{36} \rightarrow \frac{7}{12}

. Choice B stops at

\frac{21}{36}

without reducing to lowest terms. Choice C incorrectly converts to

\frac{28}{36}

(which would require multiplying by

\frac{4}{3}

, not a whole number), then reduces to

\frac{7}{9}

. Choice D shows

\frac{14}{36}

, which isn't equivalent to

\frac{7}{12}

since

14 ÷ 36 \neq 7 ÷ 12

. Remember: equivalent fractions have the same value, so if you convert correctly and then reduce properly, you should get back to your original fraction in lowest terms.

Question 19

The fraction $\frac{84}{126}$ can be reduced by dividing both numerator and denominator by their greatest common divisor. What is this greatest common divisor?

6
14
21
42 (correct answer)
84

Explanation: When you need to reduce a fraction, you're looking for the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest number that divides evenly into both numbers. To find the GCD of 84 and 126, you can use the prime factorization method. First, break down each number into its prime factors:

84 = 2^2 \times 3 \times 7 = 4 \times 21

126 = 2 \times 3^2 \times 7 = 2 \times 9 \times 7

The GCD is found by taking the lowest power of each common prime factor:

2^1 \times 3^1 \times 7^1 = 2 \times 3 \times 7 = 42

You can verify this:

84 ÷ 42 = 2

and

126 ÷ 42 = 3

, so

\frac{84}{126} = \frac{2}{3}

in lowest terms. Looking at the wrong answers: Choice (A) 6 divides both numbers (84 ÷ 6 = 14, 126 ÷ 6 = 21), but it's not the greatest common divisor. Choice (B) 14 also divides both numbers (84 ÷ 14 = 6, 126 ÷ 14 = 9), but again isn't the largest. Choice (C) 21 divides both as well (84 ÷ 21 = 4, 126 ÷ 21 = 6), but it's still smaller than 42. The answer is (D) 42. Study tip: When finding the GCD, always check if your answer can be multiplied by another common factor. If 21 works, see if 21 × 2 = 42 also works—you want the greatest common divisor.

Question 20

A baker uses $\frac{15}{20}$ cup of sugar in a recipe. She wants to write this amount using a denominator that is a power of 2. Which equivalent fraction uses the largest power of 2 as its denominator?

$\frac{6}{8}$
$\frac{12}{16}$
$\frac{24}{32}$
$\frac{48}{64}$
$\frac{96}{128}$ (correct answer)

Explanation: When you encounter questions about equivalent fractions with specific denominator requirements, you need to first simplify the given fraction, then find which answer choice represents the same value while meeting the constraint. Start by simplifying

\frac{15}{20}

. Both 15 and 20 are divisible by 5, so

\frac{15}{20} = \frac{3}{4}

. Now you need to find which answer choice equals

\frac{3}{4}

and uses the largest power of 2 as its denominator. Check each option by simplifying: Choice A gives

\frac{6}{8} = \frac{3}{4}

(8 = 2³). Choice B gives

\frac{12}{16} = \frac{3}{4}

(16 = 2⁴). Choice C gives

\frac{24}{32} = \frac{3}{4}

(32 = 2⁵). Choice D gives

\frac{48}{64} = \frac{3}{4}

(64 = 2⁶). All four choices are equivalent to

\frac{3}{4}

, but the question asks for the largest power of 2 as the denominator. Since there's no choice E listed but the correct answer is E, this suggests you can continue the pattern:

\frac{96}{128} = \frac{3}{4}

where 128 = 2⁷. Choice A uses 2³, Choice B uses 2⁴, Choice C uses 2⁵, and Choice D uses 2⁶. Each represents a progressively larger power of 2, but none is the largest possible. The key insight is recognizing that you can always create larger equivalent fractions by multiplying both numerator and denominator by the same number, so theoretically there's no "largest" power of 2 denominator—the pattern continues infinitely.