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Lesson 4 of 7

Proportional Reasoning-2 · Lesson 4 of 7

Dividing a Whole in a Given Ratio

“Learn how to split a fixed total into two or more proportional parts and apply the method to mixtures, time, money and geometry.”

Objectives
  • Divide a total quantity into parts that follow a given ratio.
  • Explain why adding the ratio terms gives the total number of equal parts.
  • Use a general formula for dividing a whole among several terms.
  • Apply ratio division to mixtures, schedules, money and triangle angles.
  • Reason about triangle side ratios and the triangle inequality.

From Ratio Parts to Actual Amounts

When a whole amount must be divided in a given ratio, the ratio tells us how many equal-sized parts belong to each share. The first step is to count the total number of ratio-parts. Then we find the value of one part and build each share from it.

Example — Dividing 12 in the ratio 2 : 1

Problem
Divide 12 in the ratio 2 : 1.

  1. 1.Add the ratio terms: 2 + 1 = 3 total parts.
  2. 2.One part = 12 ÷ 3 = 4.
  3. 3.First share = 2 × 4 = 8.
  4. 4.Second share = 1 × 4 = 4.
  5. 5.Check: 8 + 4 = 12, and 8 : 4 simplifies to 2 : 1.

The General Method

The same reasoning works for any number of terms. If a total x is divided in the ratio a : b : c : ..., then a + b + c + ... is the total number of ratio-parts. Each share is the whole multiplied by that term's fraction of the total parts.

Dividing a quantity in a multi-term ratioLaTeX
The shares should add back to x.
Whole = 10 equal parts2 parts3 parts5 parts
Dividing a whole in the ratio 2 : 3 : 5— The bar has 10 equal ratio-parts. The three groups contain 2, 3 and 5 parts.
Example — Dividing 110 units of concrete

Problem
Cement, sand and gravel are mixed in the ratio 1 : 1.5 : 3. How should 110 units of concrete be divided among the three materials?

  1. 1.Add the ratio terms: 1 + 1.5 + 3 = 5.5 parts.
  2. 2.One part = 110 ÷ 5.5 = 20 units.
  3. 3.Cement = 1 × 20 = 20 units.
  4. 4.Sand = 1.5 × 20 = 30 units.
  5. 5.Gravel = 3 × 20 = 60 units.
  6. 6.Check: 20 + 30 + 60 = 110.
Example — Preparing 50 mL of paint

Problem
Red, blue and white paint are mixed in the ratio 2 : 3 : 5. How much of each is needed to make 50 mL?

  1. 1.Total parts = 2 + 3 + 5 = 10.
  2. 2.Red = 50 × 2/10 = 10 mL.
  3. 3.Blue = 50 × 3/10 = 15 mL.
  4. 4.White = 50 × 5/10 = 25 mL.
  5. 5.Check: 10 + 15 + 25 = 50 mL.

Using Ratios in Geometry

Ratio division is especially useful when a fixed geometric total is known. The angles of a triangle always add to 180°, so an angle ratio tells us how that 180° must be shared.

Example — Triangle angles in the ratio 1 : 3 : 5

Problem
Find the three angles of a triangle whose angles are in the ratio 1 : 3 : 5.

  1. 1.The angle sum of a triangle is 180°.
  2. 2.Total ratio-parts = 1 + 3 + 5 = 9.
  3. 3.First angle = 180 × 1/9 = 20°.
  4. 4.Second angle = 180 × 3/9 = 60°.
  5. 5.Third angle = 180 × 5/9 = 100°.
  6. 6.Check: 20° + 60° + 100° = 180°.
ABC20°60°100°
Triangle with angles 20°, 60° and 100°— The three angles add to 180° and follow the ratio 1 : 3 : 5.

Reasoning Beyond Calculation

A ratio can describe the shape of a triangle without fixing its absolute size. Triangles with side ratio 3 : 4 : 5 can be enlarged or reduced, so they need not be congruent even though they have the same shape. Ratios can also reveal when a proposed triangle is impossible.

Triangle check

For three lengths to form a triangle, the sum of any two sides must be greater than the third side. A ratio 1 : 3 : 5 fails because 1 + 3 is less than 5, so no triangle can have sides in that ratio.

Example — Sharing 150 minutes

Problem
A practice session uses time in the ratio warm-up/cool-down : batting : bowling : fielding = 3 : 4 : 3 : 5. How should 150 minutes be divided?

  1. 1.Total parts = 3 + 4 + 3 + 5 = 15.
  2. 2.One part = 150 ÷ 15 = 10 minutes.
  3. 3.Warm-up/cool-down = 3 × 10 = 30 minutes.
  4. 4.Batting = 4 × 10 = 40 minutes.
  5. 5.Bowling = 3 × 10 = 30 minutes.
  6. 6.Fielding = 5 × 10 = 50 minutes.

Quiz

Quick check

To divide 72 in the ratio 2 : 3 : 4, what is the value of one ratio-part?

Quick check

A total of 60 is divided in the ratio 1 : 2 : 3. What is the largest share?

Quick check

Which formula gives the share corresponding to a in the ratio a : b : c when the total is x?

Quick check

The angles of a triangle are in the ratio 2 : 3 : 4. What is the smallest angle?

Quick check

Why do side lengths in the ratio 1 : 3 : 5 fail to form a triangle?

Practice Problems

Practice Problems
  1. Divide 150 minutes in the ratio 3 : 4 : 3 : 5.
  2. A library has books in the ratio Odia : Hindi : English = 3 : 2 : 1. If there are 288 Odia books, find the numbers of Hindi and English books.
  3. There are 100 coins in the ratio number of ₹10 coins : ₹5 coins : ₹2 coins : ₹1 coins = 4 : 3 : 2 : 1. Find the number of each coin and the total value of all the coins.
  4. Divide 84 in the ratio 2 : 5 : 7 and verify that the shares add to 84.
  5. Construct a triangle whose side lengths are proportional to 3 : 4 : 5. Explain why another triangle with sides 6 : 8 : 10 has the same shape but is not congruent to the first.
  6. Explain why side lengths in the ratio 1 : 3 : 5 cannot form a triangle.
  7. A sum of ₹9,600 is shared among three people in the ratio 5 : 4 : 3. Find each share.

Key Takeaways

Key Takeaways

To divide a whole in a ratio, first add the ratio terms to find the total number of parts. One part equals the whole divided by the sum of the ratio terms. Each share equals the whole multiplied by that term divided by the total of all terms. The shares must add back to the original whole. A fixed total such as 180° can be divided using the same method. A side ratio can describe shape without fixing size, and triangle side ratios must satisfy the triangle inequality.

Previous · Lesson 3

Ratios with More Than Two Terms

Next · Lesson 5

Pie Charts — Proportions of a Whole