ISEE Middle Level

Fraction Operations

Adding, subtracting, multiplying, and dividing fractions and mixed numbers — excludes percent calculations (see percent-calculations) and ratio/proportion problems (see ratios-proportions-solving)

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Imagine you and your friends order a giant pepperoni pizza! 🍕 You eat $\frac{3}{8}$ of it, and your best friend eats $\frac{1}{4}$ . How much is left? To figure that out, you need fraction operations! Working with fractions on the ISEE is like learning the rules of a fun new board game. Once you know the moves, you can win every time!

For adding and subtracting, fractions are a bit picky. They refuse to work together unless their bottom numbers (denominators) match. It’s like trying to play soccer with a basketball—you need the right equipment! So, we find a common denominator first, and then only add or subtract the top numbers.

Multiplying fractions is super chill. No matching needed! You just multiply the top numbers straight across, and then the bottom numbers straight across. Easy peasy! 😎

Dividing fractions? Time for a cool skateboard trick! We use a move called 'Keep-Change-Flip.' Keep the first fraction, change the division sign to multiplication, and flip the second fraction upside down. 🛹

On the ISEE, you'll see fractions in two ways. In the Mathematics Achievement section, you'll just do the math. In the Quantitative Reasoning section, you might compare two columns. Don't worry, just use your fraction rules and you'll crush it!

Key Formula

For division, remember Keep-Change-Flip!

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

Practice Questions

4 practice questions for ISEE Middle Level

Q1 Medium

There are 5 identical blocks of modeling clay. Sarah uses

\frac{1}{4}

of each block for a project. Mark uses

\frac{2}{5}

of each block for his project. How many blocks of clay remain in total?

1 \frac{1}{4}

blocks

1 \frac{1}{2}

blocks

1 \frac{3}{4}

blocks

3 \frac{1}{4}

blocks

Show Solution

First, find the total fraction of each block used by adding Sarah's and Mark's portions: $\frac{1}{4} + \frac{2}{5}$ . To add these fractions, find a common denominator, which is 20. $\frac{1}{4} = \frac{5}{20}$ and $\frac{2}{5} = \frac{8}{20}$ . Adding them gives $\frac{5}{20} + \frac{8}{20} = \frac{13}{20}$ . This means $\frac{13}{20}$ of each block is used, so $1 - \frac{13}{20} = \frac{7}{20}$ of each block remains. Since there are 5 blocks in total, multiply the remaining fraction by 5: $5 \times \frac{7}{20} = \frac{35}{20}$ . Simplify this fraction to $\frac{7}{4}$ , which converts to the mixed number $1 \frac{3}{4}$ .

Answer: C

Q2 Medium

What is the value of

n

in the equation

3 \frac{3}{4} \div 1 \frac{1}{8} = n

2 \frac{1}{2}

3 \frac{1}{3}

3 \frac{2}{3}

4 \frac{7}{32}

Show Solution

To divide mixed numbers, first convert them to improper fractions. $3 \frac{3}{4} = \frac{15}{4}$ and $1 \frac{1}{8} = \frac{9}{8}$ . The equation becomes $\frac{15}{4} \div \frac{9}{8} = n$ . To divide fractions, multiply the first fraction by the reciprocal of the second fraction: $\frac{15}{4} \times \frac{8}{9}$ . Multiply the numerators and denominators: $\frac{15 \times 8}{4 \times 9} = \frac{120}{36}$ . Simplify the fraction by dividing the numerator and denominator by their greatest common factor, 12: $\frac{120 \div 12}{36 \div 12} = \frac{10}{3}$ . Finally, convert the improper fraction back to a mixed number: $\frac{10}{3} = 3 \frac{1}{3}$ .

Answer: B

Q3 Medium

Of the 60% of students in a middle school who play a sport,

\frac{3}{4}

also play a musical instrument. What fraction of the students in the middle school play both a sport and a musical instrument?

\frac{9}{20}

\frac{11}{20}

\frac{3}{10}

\frac{4}{5}

Show Solution

First, convert the percentage of students who play a sport into a fraction. 60% is equal to $\frac{60}{100}$ , which simplifies to $\frac{3}{5}$ . We are told that $\frac{3}{4}$ of these students also play an instrument. To find the fraction of the total student body that does both, multiply the two fractions together: $\frac{3}{5} \times \frac{3}{4} = \frac{9}{20}$ .

Answer: A

Q4 Medium

A recipe for a batch of cookies calls for

2 \frac{1}{2}

cups of flour. If Jane wants to make exactly

\frac{1}{2}

of the recipe, but she has already put

\frac{3}{4}

of a cup of flour into the mixing bowl, how much more flour does she need to add?

\frac{1}{4}

cup

\frac{1}{2}

cup

\frac{3}{4}

cup

1 \frac{1}{4}

cups

Show Solution

First, determine the total amount of flour Jane needs for half of the recipe. Multiply the original amount of flour by $\frac{1}{2}$ : $2 \frac{1}{2} \times \frac{1}{2}$ . Convert $2 \frac{1}{2}$ to an improper fraction, which is $\frac{5}{2}$ . Multiplying gives $\frac{5}{2} \times \frac{1}{2} = \frac{5}{4}$ , or $1 \frac{1}{4}$ cups. Since she has already added $\frac{3}{4}$ of a cup, subtract this from the total amount needed: $\frac{5}{4} - \frac{3}{4} = \frac{2}{4}$ . Simplify $\frac{2}{4}$ to $\frac{1}{2}$ . Jane needs to add $\frac{1}{2}$ cup of flour.

Answer: B

Tips & Strategies

Always convert mixed numbers to improper fractions before multiplying or dividing! It saves you from making silly mistakes.
On Quantitative Comparison questions (Column A vs Column B), look for a shortcut before doing the math. Sometimes you can tell they are equal just by using rules like Keep-Change-Flip!
Remember that adding and subtracting require a common denominator, but multiplying and dividing DO NOT. Don't do extra work if you don't have to!

Common Mistakes

Watch out for adding the denominators! When you add $\frac{1}{4} + \frac{1}{4}$ , the answer is $\frac{2}{4}$ , NOT $\frac{2}{8}$ . The bottom number stays the same!
Don't forget to simplify your final answer. If you get $\frac{15}{12}$ , the ISEE will usually want you to reduce it to $\frac{5}{4}$ .

Frequently Asked Questions

Do I have to simplify my fractions on the ISEE?

Yes! The ISEE almost always puts the answer choices in their simplest form. If your answer isn't there, see if you can divide the top and bottom by the same number.

What if I forget the common denominator?

You can always just multiply the two bottom numbers together to find a common denominator! It might not be the smallest one, but it will always work.

Is there a penalty for guessing if I'm totally stuck?

Nope! The ISEE has zero guessing penalty. If a fraction problem is taking too long, pick your favorite letter and move on.

How do I remember which fraction to flip when dividing?

Always flip the SECOND fraction! Think of the phrase 'Keep-Change-Flip' reading from left to right, just like reading a book.

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