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Lesson 3 of 7

Proportional Reasoning-2 · Lesson 3 of 7

Ratios with More Than Two Terms

“Extend ratio reasoning to mixtures and situations involving three or more quantities while keeping every term in the same proportion.”

Objectives
  • Interpret ratios containing three or more terms.
  • Scale every term of a multi-term ratio by the same factor.
  • Recognise when two multi-term ratios are proportional.
  • Find missing quantities in mixtures described by multi-term ratios.
  • Use multi-term ratios in practical contexts such as paint and concrete mixtures.

One Ratio Can Compare Several Quantities

A ratio does not have to compare only two quantities. When a mixture contains several ingredients, one ratio can record how all of them are related. The order matters because each term belongs to a particular quantity.

Suppose a spice mixture uses 8 spoons of coriander seeds, 4 red chillies, 2 spoons of toor dal and 1 spoon of fenugreek seeds. In that order, the ratio is 8 : 4 : 2 : 1. This means that whenever the mixture is scaled up or down, all four quantities must change by the same factor.

Definition
Multi-term ratio

A ratio with three or more terms that compares several quantities in a fixed order, such as a : b : c or a : b : c : d.

Original: 8 : 4 : 2 : 1coriander 8chillies 4toor dal 2fenugreek 1× 1/24 : 2 : 1 : 0.5
Scaling a four-term spice ratio— Every ingredient is halved, so the flavour proportion remains unchanged.
Example — Halving a spice mixture

Problem
A spice mixture has ratio 8 : 4 : 2 : 1. If the second quantity is reduced from 4 to 2, what should the complete new ratio be?

  1. 1.The second term has been multiplied by 1/2.
  2. 2.Multiply every term by 1/2.
  3. 3.8 × 1/2 = 4, 4 × 1/2 = 2, 2 × 1/2 = 1, and 1 × 1/2 = 0.5.
  4. 4.The new ratio is 4 : 2 : 1 : 0.5, which is proportional to the original ratio.

Recognising Equivalent Multi-Term Ratios

Two multi-term ratios are proportional when the same multiplier connects each corresponding pair of terms. Instead of checking only one pair, we must make sure the scale factor works for every term.

Equivalent multi-term ratiosLaTeX
All corresponding ratios must have the same value.
Example — Checking four terms

Problem
Are 12 : 6 : 3 : 1.5 and 8 : 4 : 2 : 1 proportional?

  1. 1.Compare corresponding terms: 12/8 = 1.5.
  2. 2.Check the others: 6/4 = 1.5, 3/2 = 1.5, and 1.5/1 = 1.5.
  3. 3.The same factor works for every term, so the ratios are proportional.

Using Parts to Find Missing Quantities

A useful way to solve a mixture problem is to first find the size of one ratio-part. If one known quantity corresponds to several parts, divide its actual amount by its number of parts. Then multiply the one-part value by the other terms.

Example — Mixing purple paint

Problem
Red, blue and white paint are mixed in the ratio 2 : 3 : 5. If the white paint is 10 L, how much red and blue paint are needed?

  1. 1.White corresponds to 5 parts, and 5 parts = 10 L.
  2. 2.One part = 10 ÷ 5 = 2 L.
  3. 3.Red is 2 parts, so red = 2 × 2 = 4 L.
  4. 4.Blue is 3 parts, so blue = 3 × 2 = 6 L.
  5. 5.The complete mixture is 4 + 6 + 10 = 20 L.
Example — Scaling a concrete mixture

Problem
Cement, sand and gravel are mixed in the ratio 1 : 1.5 : 3. If 3 bags of cement are used, how many bags of sand and gravel are needed, and what is the total mixture?

  1. 1.Cement changes from 1 part to 3 bags, so the scale factor is 3.
  2. 2.Sand = 1.5 × 3 = 4.5 bags.
  3. 3.Gravel = 3 × 3 = 9 bags.
  4. 4.Total mixture = 3 + 4.5 + 9 = 16.5 bags.
Common mistake

Do not add or subtract the same number from each term to make an equivalent ratio. Equivalent ratios come from multiplying or dividing every term by the same factor.

Quiz

Quick check

Which ratio is proportional to 8 : 4 : 2 : 1?

Quick check

A mixture has ratio A : B : C = 2 : 3 : 5. If C is 20 units, what is one part worth?

Quick check

In the ratio 3 : 4 : 7, the second term represents 20 kg. What does the first term represent?

Quick check

Which statement must be true for two four-term ratios to be proportional?

Quick check

A 2 : 3 : 5 paint ratio is scaled so that red becomes 8 L. How much white paint is needed?

Practice Problems

Practice Problems
  1. Scale 6 : 3 : 2 by a factor of 4.
  2. Decide whether 9 : 6 : 3 and 15 : 10 : 5 are proportional. Show the common scale factor if they are.
  3. Juice concentrate, water and ice are used in the ratio 1 : 4 : 2. If 12 cups of water are used, find the amounts of concentrate and ice.
  4. A colour mixture uses red, blue and white in the ratio 2 : 3 : 5. If 15 L of blue paint is available, find the matching amounts of red and white paint and the total volume.
  5. Cement, sand and gravel are in the ratio 1 : 1.5 : 3. If 7.5 bags of sand are used, find the amounts of cement and gravel.
  6. Write a four-term ratio proportional to 5 : 3 : 2 : 1 in which the first term is 20.

Key Takeaways

Key Takeaways

Ratios can compare three, four or more quantities at once. The order of terms matters because each term refers to a specific quantity. Equivalent multi-term ratios are created by multiplying or dividing every term by the same factor. A known term can be used to find the size of one ratio-part. Once one part is known, every other quantity follows from its number of parts.

Previous · Lesson 2

Ratios in Maps

Next · Lesson 4

Dividing a Whole in a Given Ratio