Ratio Word Problems with Step-by-Step Solutions

Grade: 6-7 | Topic: Arithmetic

What You Will Learn

After this lesson you will be able to solve common types of ratio word problems including sharing a quantity in a given ratio, finding missing amounts, scaling recipes, and comparing quantities. You will master the "parts method" — a reliable strategy that works on every ratio word problem.

Theory

The "parts" method

Most ratio word problems ask you to divide a total into parts described by a ratio. The strategy is always the same:

1. Add the ratio parts to find the total number of parts.
2. Divide the actual total by the number of parts to find the value of one part.
3. Multiply to find each share.

For a ratio $a : b$ sharing a total $T$:

$\text{Total parts} = a + b$

$\text{Value of one part} = \frac{T}{a + b}$

$\text{First share} = a \times \frac{T}{a + b} \qquad \text{Second share} = b \times \frac{T}{a + b}$

When only part of the total is known

Sometimes the problem gives you one share instead of the total. In that case:

1. Set up the ratio parts.
2. Use the known share to find the value of one part.
3. Multiply to find the unknown share(s) or the total.

For example, if the ratio is $3 : 5$ and the first share is 24:

$3 \text{ parts} = 24 \implies 1 \text{ part} = 8$

$\text{Second share} = 5 \times 8 = 40$

When the difference is known

If you are told the difference between two shares:

$\text{Difference in parts} = |a - b|$

$\text{Value of one part} = \frac{\text{difference}}{|a - b|}$

Then multiply each ratio part by the value of one part to find the actual amounts.

Worked Examples

Example 1: Sharing money in a ratio (easy)

Problem: Sarah and Tom share $60 in the ratio $2 : 3$. How much does each person get?

Step 1: Total parts = $2 + 3 = 5$.

Step 2: Value of one part = $\frac{60}{5} = 12$.

Step 3: Calculate each share.

$\text{Sarah} = 2 \times 12 = \$24$

$\text{Tom} = 3 \times 12 = \$36$

Answer: Sarah gets $24 and Tom gets $36.

Check: $24 + 36 = 60$ ✓ and $24 : 36 = 2 : 3$ ✓

Example 2: Three-way split (medium)

Problem: Three friends share 180 stickers in the ratio $2 : 3 : 4$. How many stickers does each friend get?

Step 1: Total parts = $2 + 3 + 4 = 9$.

Step 2: Value of one part = $\frac{180}{9} = 20$.

Step 3: Calculate each share.

$\text{Friend A} = 2 \times 20 = 40$

$\text{Friend B} = 3 \times 20 = 60$

$\text{Friend C} = 4 \times 20 = 80$

Answer: The friends get 40, 60, and 80 stickers respectively.

Check: $40 + 60 + 80 = 180$ ✓

Example 3: Finding the total from one share (medium)

Problem: The ratio of boys to girls in a club is $5 : 3$. If there are 20 boys, how many girls are there and how many members are in the club?

Step 1: Boys represent 5 parts, and 5 parts = 20.

$\text{Value of one part} = \frac{20}{5} = 4$

Step 2: Girls represent 3 parts.

$\text{Girls} = 3 \times 4 = 12$

Step 3: Total members.

$\text{Total} = 20 + 12 = 32$

Answer: There are 12 girls and 32 members total.

Example 4: Using the difference (medium)

Problem: Two brothers share an inheritance in the ratio $7 : 4$. The older brother receives $1,500 more than the younger brother. How much does each receive?

Step 1: Difference in parts = $7 - 4 = 3$ parts.

Step 2: 3 parts = $1,500, so one part = $\frac{1500}{3} = 500$.

Step 3: Calculate each share.

$\text{Older brother} = 7 \times 500 = \$3{,}500$

$\text{Younger brother} = 4 \times 500 = \$2{,}000$

Answer: The older brother gets $3,500 and the younger brother gets $2,000.

Check: $3500 - 2000 = 1500$ ✓ and $3500 : 2000 = 7 : 4$ ✓

Example 5: Recipe scaling with a ratio (challenging)

Problem: A fruit punch recipe mixes orange juice, apple juice, and water in the ratio $3 : 2 : 5$. If you want to make 4 liters of punch, how many milliliters of each ingredient do you need?

Step 1: Convert to milliliters: 4 liters = 4000 mL.

Step 2: Total parts = $3 + 2 + 5 = 10$.

Step 3: Value of one part = $\frac{4000}{10} = 400$ mL.

Step 4: Calculate each ingredient.

$\text{Orange juice} = 3 \times 400 = 1200 \text{ mL}$

$\text{Apple juice} = 2 \times 400 = 800 \text{ mL}$

$\text{Water} = 5 \times 400 = 2000 \text{ mL}$

Answer: 1200 mL of orange juice, 800 mL of apple juice, and 2000 mL of water.

Check: $1200 + 800 + 2000 = 4000$ mL ✓

Common Mistakes

Mistake 1: Using the ratio numbers as the actual amounts

❌ "The ratio is $2 : 3$, so Sarah gets $2 and Tom gets $3."

✅ The ratio tells you the relative sizes, not the actual amounts. You need to calculate the value of one part first.

Why this matters: Ratio numbers represent parts, not quantities. A ratio of $2 : 3$ means "for every 2 one person gets, the other gets 3" — the actual amounts depend on the total.

Mistake 2: Adding instead of multiplying to find each share

❌ One part = 12, so Sarah's share = $12 + 2 = 14$

✅ Sarah's share = $12 \times 2 = 24$

Why this matters: Each person gets their number of parts times the value of one part. Addition gives a meaningless result.

Mistake 3: Using the total instead of the difference

❌ Problem says "A has $100 more than B" with ratio $5 : 3$. Student does $\frac{100}{8} = 12.5$ per part.

✅ The difference is $5 - 3 = 2$ parts. One part = $\frac{100}{2} = 50$.

Why this matters: When the problem gives a difference (not a total), divide by the difference in parts, not the sum. Read the problem carefully to identify what number represents.

Practice Problems

Try these on your own before checking the answers:

1. Divide 72 in the ratio $4 : 5$.
2. A painter mixes red and white paint in the ratio $1 : 4$. How much of each color is needed to make 15 liters?
3. The ratio of cats to dogs at a shelter is $3 : 7$. If there are 21 cats, how many dogs are there?
4. Two workers split their earnings in the ratio $5 : 3$. If the first worker earned $200 more than the second, how much did each earn?
5. A map uses a scale of $1 : 50{,}000$. If two towns are 8 cm apart on the map, what is the real distance in km?

Click to see answers

1. Total parts = 9. One part = $\frac{72}{9} = 8$. Shares: $4 \times 8 = 32$ and $5 \times 8 = 40$.
2. Total parts = $1 + 4 = 5$. One part = $\frac{15}{5} = 3$ L. Red: 3 L, White: 12 L.
3. 3 parts = 21, so one part = 7. Dogs = $7 \times 7 = 49$.
4. Difference in parts = $5 - 3 = 2$. One part = $\frac{200}{2} = 100$. First: $500, Second: $300.
5. $8 \text{ cm} \times 50{,}000 = 400{,}000 \text{ cm} = 4 \text{ km}$.

Summary

• Use the parts method: add ratio parts, divide the total to find one part's value, then multiply.
• When only one share is known, use it to find the value of one part, then scale to the others.
• When the difference between shares is given, divide by the difference in ratio parts (not the sum).
• Always check that your shares add up to the total and simplify to the original ratio.

Related Topics

• Ratios and Proportions — Complete Guide with Examples
• How to Simplify Ratios — Methods and Examples
• Unit Rate — How to Find and Compare Unit Rates

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