Units and Parts

When two quantities are divided differently, each whole needs its own part scale. A stated equal remainder or known amount can connect the scales. That is the heart of units and parts: keep the labels separate until the story gives you a real equality.

Parents: ask your child to name the two independent wholes before drawing any bars. Then ask what the question says is equal.

By the end you will be able to give each whole a separate label, value each scale from a known part, compare the original wholes, and check the fractions in the story.

When this method helps

Use units and parts when two wholes are partitioned differently. If two unknown fractional portions are equal, making units equal may be more direct. Do not call two parts equal just because both are written as 1u: the labels can describe different scales.

Ravi spent $\frac{2}{5}$ of his money. Nina spent $\frac{1}{4}$ of hers. Each had $72 left. How much more money did Ravi have than Nina at first?

Give each whole its own label

Ravi spent $\frac{2}{5}$, so $\frac{3}{5}$ remained. Name $\frac{1}{5}$ of Ravi's money r. Nina spent $\frac{1}{4}$, so $\frac{3}{4}$ remained. Name $\frac{1}{4}$ of Nina's money n. The labels are deliberately different because fifths and quarters are different partitions.

Write the equal remainders

$$3r = \$72$$

$$3n = \$72$$

$$r = \$72 \div 3 = \$24$$

$$n = \$72 \div 3 = \$24$$

Find each original whole

Now value the whole bars. Ravi's money is five r parts. Nina's money is four n parts. Because both labels happen to be $24 here, the original totals can be compared.

Work back to the original totals

$$\text{Ravi} = 5r = 5 \times \$24 = \$120$$

$$\text{Nina} = 4n = 4 \times \$24 = \$96$$

$$\text{Difference} = \$120 - \$96 = \$24$$

🔬

Units and parts: connect the equal remainders

🤔 Predict first: What connects the two independent scales?

The equal known remaindersThe labels themselves

Check the story

$$\frac{2}{5} \times \$120 = \$48$$

$$\$120 - \$48 = \$72$$

$$\frac{1}{4} \times \$96 = \$24$$

$$\$96 - \$24 = \$72$$

$$\text{Original difference} = \$120 - \$96 = \$24$$

Why the scales stay separate

$\frac{1}{5}$ of Ravi's money and $\frac{1}{4}$ of Nina's money are not automatically the same amount. In this question, $\frac{3}{5}$ and $\frac{3}{4}$ both equal $72, so r and n work out to the same value. That equality comes from the stated remainder, not from the letters.

Watch out — easily mixed up

Quick recap

Graduated practice

Open each solution only after you have identified the known part or remainder for both wholes.

1. Compare paper lengths

Ada used $\frac{1}{3}$ of her paper strip. Ben used $\frac{2}{5}$ of his. Each had 60 cm left. Whose strip was longer at first, and by how much?

Show solution

Ada keeps two parts, while Ben keeps three. Use a for Ada's $\frac{1}{3}$ part and b for Ben's $\frac{1}{5}$ part.

Practice 1

$$2a = 60\text{ cm}$$

$$a = 60 \div 2 = 30\text{ cm}$$

$$\text{Ada} = 3 \times 30 = 90\text{ cm}$$

$$3b = 60\text{ cm}$$

$$b = 60 \div 3 = 20\text{ cm}$$

$$\text{Ben} = 5 \times 20 = 100\text{ cm}$$

$$\text{Difference} = 100 - 90 = 10\text{ cm}$$

Ben's strip was 10 cm longer. Check that each used fraction leaves 60 cm.

2. Equal used amounts

Cara used $\frac{3}{8}$ of a ribbon and Dev used $\frac{2}{5}$ of another ribbon. Each used 30 cm. Find both original lengths.

Show solution

The known amount is the used portion, so value the two scales independently.

Practice 2

$$3c = 30\text{ cm}$$

$$c = 30 \div 3 = 10\text{ cm}$$

$$\text{Cara} = 8 \times 10 = 80\text{ cm}$$

$$2d = 30\text{ cm}$$

$$d = 30 \div 2 = 15\text{ cm}$$

$$\text{Dev} = 5 \times 15 = 75\text{ cm}$$

Cara's ribbon was 80 cm and Dev's was 75 cm. Check that both named used portions are 30 cm.

3. Unequal known remainders

Eli spent $\frac{1}{4}$ of his money and had $90 left. Farah spent $\frac{2}{5}$ of hers and had $72 left. Who had more at first?

Show solution

Eli keeps $\frac{3}{4}$; Farah keeps $\frac{3}{5}$. The two unit labels remain separate even though the final totals coincide.

Practice 3

$$3e = \$90$$

$$e = \$90 \div 3 = \$30$$

$$\text{Eli} = 4 \times \$30 = \$120$$

$$3f = \$72$$

$$f = \$72 \div 3 = \$24$$

$$\text{Farah} = 5 \times \$24 = \$120$$

$$\text{Difference} = \$120 - \$120 = \$0$$

They had equal amounts at first: $120 each.

🎯 Mastery check

Answer all 7 — your progress is saved on this device.

1. When is units-and-parts reasoning useful?

   Two wholes are partitioned differentlyThe same whole has two equivalent fractionsA transfer keeps one total fixed

2. Ravi spent $\frac{2}{5}$ of his money. How many parts remained?

   2 parts3 parts5 parts

3. Which equations use Ravi and Nina's equal $72 remainders?

   3r = $72 and 3n = $725r = $72 and 4n = $722r = $72 and 1n = $72

4. What was Ravi's original amount?

   $96$120$144

5. What was Nina's original amount?

   $72$96$120

6. Why are Ravi's and Nina's parts named r and n?

   Fifths and quarters are different partitionsThe letters must be different in every equationNina's parts are always smaller

7. Which check confirms the main answer?

   Both people have $72 left after their stated spendingThe denominators are equalBoth original totals are equal

See my score