ISEE Lower Level

Comparing & Ordering

Q: Why does the ISEE mix fractions, decimals, and percents in one question?

The ISEE wants to test your 'number sense.' They want to see if you understand that `\frac{1}{2}`, `0.5`, and `50\%` are just three different costumes for the exact same amount!

Q: Do I lose points if I guess on a Quantitative Comparison question?

Nope! The ISEE has no guessing penalty. If you are totally stuck comparing two numbers, eliminate any answers you know are wrong and take your best guess.

Q: Should I convert everything to fractions or to decimals?

Usually, converting everything to decimals or percents is faster and less prone to errors. Finding common denominators for fractions can take up too much time on a timed test.

Q: What if I don't know the decimal for a weird fraction like `\frac{5}{9}`?

You don't always have to! Use benchmark fractions. You know `\frac{1}{2}` is `\frac{4.5}{9}`, so `\frac{5}{9}` is just a little bit more than `50\%`.

Comparing fractions, decimals, and percents on a number line

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Imagine you are at an epic pizza party 🍕 and you are trying to figure out who ate the most. Your friend Alex says, 'I ate $\frac{1}{2}$ of a pizza!' Your friend Bailey says, 'I ate $0.45$ of a pizza!' And your friend Charlie brags, 'I ate $40%$ of a pizza!' Who is the ultimate pizza champion?

Right now, it is super hard to tell because everyone is speaking a different math language! This is exactly what happens on the ISEE. The test makers love to mix fractions, decimals, and percents in the same question to see if you can figure out which number is the biggest, which is the smallest, or if they are perfectly equal.

The secret trick? Make them all wear the same costume! 🦸‍♂️ If you change all the numbers into decimals or all into percents, it becomes incredibly easy to line them up and compare them. Think of it like making everyone speak the exact same language. For our pizza party, $\frac{1}{2}$ is $0.50$ , $0.45$ is just $0.45$ , and $40%$ is $0.40$ . Suddenly, it is obvious that Alex, who ate $0.50$ , is the winner!

On the ISEE, you will also see 'Quantitative Comparison' questions where you have to weigh Column A against Column B. By using fun tricks like the Butterfly Method (cross-multiplying fractions) or plugging in real numbers, you can conquer these comparing questions in seconds!

Key Formula

To compare two fractions quickly without finding a common denominator, use the Butterfly Method (cross-multiply)! For

\frac{a}{b}

and

\frac{c}{d}

, multiply

a \cdot d

(write it above the first fraction) and

b \cdot c

(write it above the second). The larger product points to the larger fraction!

Practice Questions

4 practice questions for ISEE Lower Level

Q1 Easy

Which is the largest fraction?

\frac{3}{7}

\frac{4}{9}

\frac{5}{11}

\frac{6}{13}

Show Solution

To find the largest fraction, you can compare how far each is from $\frac{1}{2}$ . The difference between $\frac{1}{2}$ and $\frac{3}{7}$ is $\frac{1}{14}$ . The difference between $\frac{1}{2}$ and $\frac{4}{9}$ is $\frac{1}{18}$ . The difference between $\frac{1}{2}$ and $\frac{5}{11}$ is $\frac{1}{22}$ . The difference between $\frac{1}{2}$ and $\frac{6}{13}$ is $\frac{1}{26}$ . Since $\frac{1}{26}$ is the smallest difference, $\frac{6}{13}$ is closest to $\frac{1}{2}$ . Because all of these fractions are slightly less than $\frac{1}{2}$ , the one closest to $\frac{1}{2}$ is the largest.

Answer: D

Q2 Easy

Which fraction is between

\frac{1}{4}

and

\frac{2}{3}

\frac{1}{5}

\frac{3}{4}

\frac{5}{8}

\frac{7}{10}

Show Solution

Convert the fractions to decimals to easily compare them. $\frac{1}{4} = 0.25$ and $\frac{2}{3}$ is approximately $0.67$ . Let's evaluate the choices:

(A) $\frac{1}{5} = 0.20$ , which is less than 0.25.
(B) $\frac{3}{4} = 0.75$ , which is greater than 0.67.
(C) $\frac{5}{8} = 0.625$ , which is between 0.25 and 0.67.
(D) $\frac{7}{10} = 0.70$ , which is greater than 0.67.
Therefore, $\frac{5}{8}$ is the only fraction between $\frac{1}{4}$ and $\frac{2}{3}$ .

Answer: C

Q3 Easy

Sarah, John, and Emily are measuring the lengths of their pet caterpillars. Sarah's is 3.45 cm, John's is 3.5 cm, and Emily's is 3.05 cm. Which list shows the lengths in order from shortest to longest?

A 3.05, 3.45, 3.5

B 3.05, 3.5, 3.45

C 3.45, 3.05, 3.5

D 3.5, 3.45, 3.05

Show Solution

To order the decimals, compare the digits from left to right. All three numbers have a 3 in the ones place. Looking at the tenths place: Emily's has a 0, Sarah's has a 4, and John's has a 5. Since 0 < 4 < 5, the numbers in order from shortest to longest are 3.05, 3.45, and 3.5.

Answer: A

Q4 Easy

Which of the following inequalities is true?

0.4 > \frac{1}{2}

\frac{3}{5} < 0.5

0.75 > \frac{4}{5}

\frac{2}{3} > 0.6

Show Solution

Convert the fractions to decimals to make the comparison easier.

(A) $\frac{1}{2} = 0.5$ , and 0.4 is not greater than 0.5.
(B) $\frac{3}{5} = 0.6$ , and 0.6 is not less than 0.5.
(C) $\frac{4}{5} = 0.8$ , and 0.75 is not greater than 0.8.
(D) $\frac{2}{3}$ is approximately 0.667, and 0.667 is greater than 0.6. This inequality is true.

Answer: D

Tips & Strategies

When comparing mixed numbers, decimals, and percents, convert them all to the same format. Decimals are usually the easiest to line up and compare!
Use benchmark fractions! If you know $\frac{1}{2}$ is $50%$ , you can easily figure out that $\frac{3}{7}$ is less than $50%$ because 3 is less than half of 7.
On Quantitative Comparison questions, if both columns are just fractions, don't waste time finding a common denominator. Just cross-multiply (the Butterfly Method) to save precious seconds!

Common Mistakes

Watch out for fractions between 0 and 1! Normally, squaring a number makes it bigger (like $3^{2} = 9$ ), but squaring a proper fraction makes it smaller (like $(\frac{1}{2})^{2} = \frac{1}{4}$ ).
Don't forget that $0.4$ is the same as $0.40$ and $40%$ . Students often confuse $0.4$ with $4%$ . Always add that invisible zero to help you compare decimals properly!

Frequently Asked Questions

Why does the ISEE mix fractions, decimals, and percents in one question?

The ISEE wants to test your 'number sense.' They want to see if you understand that $\frac{1}{2}$ , $0.5$ , and $50%$ are just three different costumes for the exact same amount!

Do I lose points if I guess on a Quantitative Comparison question?

Nope! The ISEE has no guessing penalty. If you are totally stuck comparing two numbers, eliminate any answers you know are wrong and take your best guess.

Should I convert everything to fractions or to decimals?

Usually, converting everything to decimals or percents is faster and less prone to errors. Finding common denominators for fractions can take up too much time on a timed test.

What if I don't know the decimal for a weird fraction like

\frac{5}{9}

You don't always have to! Use benchmark fractions. You know $\frac{1}{2}$ is $\frac{4.5}{9}$ , so $\frac{5}{9}$ is just a little bit more than $50%$ .

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