ISEE Upper Level

Comparing & Ordering

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 Upper Level

Q1 Hard

Which of the following sets of numbers is ordered from least to greatest?

0.62

\frac{5}{8}

62.8%

\frac{2}{3}

62.8%

0.62

\frac{5}{8}

\frac{2}{3}

\frac{2}{3}

62.8%

\frac{5}{8}

0.62

0.62

\frac{2}{3}

\frac{5}{8}

62.8%

Show Solution

To compare and order these values, convert them all to a common format, such as decimals:

• $\frac{5}{8} = 5 \div 8 = 0.625$
• $0.62 = 0.62$
• $62.8% = 62.8 \div 100 = 0.628$
• $\frac{2}{3} = 2 \div 3 = 0.666...$ (approximately $0.667$ )
Now, order the decimal values from least to greatest:
$0.62$ (from $0.62$ )
$0.625$ (from $\frac{5}{8}$ )
$0.628$ (from $62.8%$ )
$0.666...$ (from $\frac{2}{3}$ )
Therefore, the correct order from least to greatest is $0.62$ , $\frac{5}{8}$ , $62.8%$ , $\frac{2}{3}$ .

Answer: A

Q2 Hard

Four students are comparing their progress on a math test. Alex completed

\frac{13}{16}

of the problems, Brenda completed

81.5%

, Carlos completed

0.8125

, and Diane completed

\frac{65}{80}

. Who completed the greatest proportion of the problems?

A Alex

B Brenda

C Carlos

D Diane

Show Solution

To determine who completed the greatest proportion, convert all proportions to a common format, such as decimals:

• Alex: $\frac{13}{16} = 13 \div 16 = 0.8125$
• Brenda: $81.5% = 81.5 \div 100 = 0.815$
• Carlos: $0.8125$
• Diane: $\frac{65}{80}$ . This fraction can be simplified by dividing both the numerator and the denominator by 5: $\frac{65 \div 5}{80 \div 5} = \frac{13}{16}$ . So, Diane completed $\frac{13}{16}$ , which is $0.8125$ in decimal form.
Now, compare the decimal values:
• Alex: $0.8125$
• Brenda: $0.815$
• Carlos: $0.8125$
• Diane: $0.8125$
Comparing these values, $0.815$ is the greatest. Therefore, Brenda completed the greatest proportion of the problems.

Answer: B

Q3 Hard

Which of the following values does NOT lie strictly between

\frac{4}{7}

and

0.58

on a number line?

57.2%

\frac{11}{19}

0.5801

0.579

Show Solution

First, convert the boundary values of the interval to decimals to better understand the range:

• $\frac{4}{7} \approx 0.571428...$
• $0.58$
The interval is approximately (0.571428, 0.58). Now, convert each answer choice to a decimal and check if it falls within this interval:
• A. $57.2%$ : $57.2 \div 100 = 0.572$ . This value is greater than $0.571428$ and less than $0.58$ , so it lies strictly within the interval.
• B. $\frac{11}{19}$ : $11 \div 19 \approx 0.578947...$ . This value is greater than $0.571428$ and less than $0.58$ , so it lies strictly within the interval.
• C. $0.5801$ : This value is greater than $0.58$ , so it does NOT lie strictly between $\frac{4}{7}$ and $0.58$ .
• D. $0.579$ : This value is greater than $0.571428$ and less than $0.58$ , so it lies strictly within the interval.
Therefore, $0.5801$ is the value that does NOT lie strictly between $\frac{4}{7}$ and $0.58$ .

Answer: C

Q4 Hard

Three different investments showed varying returns last month. Investment P had a return of

- \frac{1}{2} %

, Investment Q had a return of

+ 0.0055

, and Investment R had a return of

- \frac{3}{500}

. Which investment had the greatest return?

A Investment P

B Investment Q

C Investment R

D Investments P and R had the same return.

Show Solution

To compare the returns, convert all values to a common format, such as decimals:

• Investment P: $- \frac{1}{2} % = - 0.5%$ . To convert a percentage to a decimal, divide by 100: $- 0.5 \div 100 = - 0.005$ .
• Investment Q: $+ 0.0055$ . This is already in decimal form.
• Investment R: $- \frac{3}{500}$ . To convert this fraction to a decimal, divide 3 by 500: $- 3 \div 500 = - 0.006$ .
Now, compare the decimal returns:
• Investment P: $- 0.005$
• Investment Q: $+ 0.0055$
• Investment R: $- 0.006$
To find the greatest return, we look for the largest number. Positive numbers are always greater than negative numbers. Among the negative numbers, $- 0.005$ is greater than $- 0.006$ . However, $+ 0.0055$ is a positive number and thus the greatest of all three.
Ordering them from least to greatest:
$- 0.006$ (Investment R)
$- 0.005$ (Investment P)
$+ 0.0055$ (Investment Q)
Therefore, Investment Q had the greatest return.

Answer: B

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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