ISEE Middle 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 Middle Level

Q1 Medium

Which of the following lists the numbers

\frac{3}{5}

0.62

58%

, and

\frac{2}{3}

in order from least to greatest?

\frac{3}{5}, 0.62, 58%, \frac{2}{3}

58%, \frac{3}{5}, 0.62, \frac{2}{3}

\frac{2}{3}, 0.62, \frac{3}{5}, 58%

58%, 0.62, \frac{2}{3}, \frac{3}{5}

Show Solution

To compare and order these numbers, it's best to convert them all to the same format, such as decimals.
1. Convert $\frac{3}{5}$ to a decimal: $3 \div 5 = 0.6$
2. The decimal $0.62$ is already in decimal form.
3. Convert $58%$ to a decimal: $58 \div 100 = 0.58$
4. Convert $\frac{2}{3}$ to a decimal: $2 \div 3 = 0.666...$ (We can use $0.67$ for comparison, or $0.66$ to maintain more precision if needed, but $0.666...$ is exact.)

Now, compare the decimal values:
$0.58$ ( $58%$ )
$0.60$ ( $\frac{3}{5}$ )
$0.62$
$0.666...$ ( $\frac{2}{3}$ )
Ordering them from least to greatest gives:
$0.58$ ( $58%$ )
$0.60$ ( $\frac{3}{5}$ )
$0.62$
$0.666...$ ( $\frac{2}{3}$ )
Therefore, the correct order is $58%, \frac{3}{5}, 0.62, \frac{2}{3}$ .

Answer: B

Q2 Medium

Alex, Ben, and Chloe were each assigned the same amount of homework. Alex completed

\frac{7}{8}

of his homework, Ben completed

0.85

of his homework, and Chloe completed

82%

of her homework. Who completed the most homework?

A Alex

B Ben

C Chloe

D They all completed the same amount.

Show Solution

To determine who completed the most homework, we need to compare the portions completed by each person. It's easiest to compare them by converting all values to decimals.
1. Alex's progress: $\frac{7}{8}$

To convert the fraction $\frac{7}{8}$ to a decimal, divide 7 by 8: $7 \div 8 = 0.875$ .
2. Ben's progress: $0.85$

This value is already in decimal form.
3. Chloe's progress: $82%$

To convert a percentage to a decimal, divide by 100: $82 \div 100 = 0.82$ .
Now, let's compare the decimal values:
Alex: $0.875$
Ben: $0.850$ (adding a zero to match decimal places for easier comparison)
Chloe: $0.820$ (adding a zero to match decimal places for easier comparison)
Comparing $0.875$ , $0.850$ , and $0.820$ , we see that $0.875$ is the largest value.
Therefore, Alex completed the most homework.

Answer: A

Q3 Medium

Which of the following values lies between

\frac{3}{4}

and

\frac{4}{5}

on a number line?

70%

0.82

0.78

\frac{13}{20}

Show Solution

To determine which value lies between $\frac{3}{4}$ and $\frac{4}{5}$ , we first convert $\frac{3}{4}$ and $\frac{4}{5}$ to decimals or percents to establish the range.
1. Convert $\frac{3}{4}$ to a decimal: $3 \div 4 = 0.75$ .
2. Convert $\frac{4}{5}$ to a decimal: $4 \div 5 = 0.80$ .

So, we are looking for a value $x$ such that $0.75 < x < 0.80$ .
Now, let's evaluate each answer choice:
• A. $70%$ : Convert to a decimal: $70 \div 100 = 0.70$ . This value is less than $0.75$ , so it is not between the two numbers.
• B. $0.82$ : This value is greater than $0.80$ , so it is not between the two numbers.
• C. $0.78$ : This value is greater than $0.75$ and less than $0.80$ . So, $0.75 < 0.78 < 0.80$ . This is the correct answer.
• D. $\frac{13}{20}$ : Convert to a decimal: $13 \div 20 = 0.65$ . This value is less than $0.75$ , so it is not between the two numbers.
Therefore, $0.78$ is the value that lies between $\frac{3}{4}$ and $\frac{4}{5}$ on a number line.

Answer: C

Q4 Medium

A school has three seventh-grade classes. In Mrs. Smith's class,

17

out of

20

students passed the math test. In Mr. Jones's class,

83%

of the students passed. In Ms. Davis's class,

0.84

of the students passed. Which class had the highest passing rate?

A Mrs. Smith's class

B Mr. Jones's class

C Ms. Davis's class

D All classes had the same passing rate.

Show Solution

To find out which class had the highest passing rate, we need to compare the passing rates of all three classes by converting them into a common format, such as decimals or percentages.
1. Mrs. Smith's class: $17$ out of $20$ students passed.

To convert this fraction to a decimal: $17 \div 20 = 0.85$ .
As a percentage: $0.85 \times 100 = 85%$ .
2. Mr. Jones's class: $83%$ of the students passed.

As a decimal: $83 \div 100 = 0.83$ .
3. Ms. Davis's class: $0.84$ of the students passed.

As a percentage: $0.84 \times 100 = 84%$ .
Now, let's compare the passing rates:
Mrs. Smith's class: $0.85$ (or $85%$ )
Mr. Jones's class: $0.83$ (or $83%$ )
Ms. Davis's class: $0.84$ (or $84%$ )
Comparing $0.85$ , $0.83$ , and $0.84$ , the highest value is $0.85$ .
Therefore, Mrs. Smith's class had the highest passing rate.

Answer: A

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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