7.4 Problem Solving with Proportional Reasoning

Once we can test for proportion, a whole range of everyday problems opens up.

Worked Example 1

Are the ratios 5 : 6 and 60 : 72 proportional?

5 : 6 is already in simplest form. For 60 : 72, the HCF of 60 and 72 is 12, and 60 : 72 = \frac{60}{12} : \frac{72}{12} = 5 : 6. Both simplest forms agree, so the ratios are proportional: 5 : 6 :: 60 : 72.

Worked Example 2: Buttermilk Stall

Sahana prepares 8 glasses of spiced buttermilk using 14 pinches of salt. The crowd grows, and she must make 24 more glasses. To keep the taste the same, how much salt does she need for those extra glasses?

The saltiness depends only on the ratio of glasses to salt staying fixed at 8 : 14. For the 24 extra glasses we model 8 : 14 :: 24 : ? The first term jumped from 8 to 24, a factor of 24 \div 8 = 3. The second term must change by the same factor: 14 \times 3 = 42. So she should add 42 pinches of salt for the 24 extra glasses, keeping 8 : 14 :: 24 : 42.

Worked Example 3: The Boundary Fence

Ravi puts up a 75 ft stretch of fencing using 5 bags of concrete; Sita puts up a 45 ft stretch using 3 bags. Ravi worries Sita’s stretch is flimsier. Is he right?

Compare each person’s ratio of fence length to bags of concrete: \text{Ravi: } 75 : 5 = 15 : 1, \qquad \text{Sita: } 45 : 3 = 15 : 1. Both reduce to 15 : 1, so the same amount of concrete went into each foot of fence. The two stretches are equally sturdy, Ravi can relax.

Worked Example 4: Ages Don’t Stay Proportional

When Diya was 4, her father was 9 times as old. What is the ratio of their ages then, and when Diya turns 16? Does it stay the same?

At age 4: the father is 36, so the ratio is 4 : 36 = 1 : 9.

Twelve years later Diya is 16 and her father is 48, giving 16 : 48 = 1 : 3.

The ratio changed, from 1 : 9 to 1 : 3, because we added the same number (12 years) to both terms. Adding (or subtracting) the same amount does not preserve a ratio; only multiplying both terms does.

Worked Example 5: Filling Proportional Blanks

Fill the blanks so each ratio is proportional to 12 : 20: \quad \underline{\ \ } : 60, \qquad 9 : \underline{\ \ }, \qquad 3 : \underline{\ \ }.

  • \underline{\ \ } : 60, the second term 60 = 3 \times 20, so the first term is 3 \times 12 = 36. Ratio: 36 : 60.
  • 9 : \underline{\ \ }, we need a factor y with 12y = 9, so y = \tfrac{9}{12} = \tfrac{3}{4}. Then the second term is 20 \times \tfrac{3}{4} = 15. Ratio: 9 : 15.
  • 3 : \underline{\ \ }, dividing 12 by its HCF with 20 (which is 4) gives 3; dividing 20 by 4 gives 5. Ratio: 3 : 5.

Masala Chai: stronger or lighter?

At her tea cart, Reshma usually steeps 12 mL of tea concentrate with 28 mL of milk, a ratio of 12 : 28 = 3 : 7. For a stronger cup she uses more concentrate per unit of milk (16 : 24 = 2 : 3, concentrate-heavy); for a lighter cup she uses less (8 : 32 = 1 : 4).

Math Talk Convert each blend to “concentrate per 100 mL of milk.” Regular 12:28 is about 43 mL per 100; strong 16:24 is about 67 mL per 100; light 8:32 is 25 mL per 100. Now you can rank the strengths even though the totals differ, that is proportional reasoning in action.

For each row below, compare the concentrate-to-milk ratio against the regular 3 : 7 and mark it stronger or lighter:

Concentrate (mL) Milk (mL) Simplest form Compared to 3:7
240 480 1 : 2 stronger
120 400 3 : 10 lighter
160 320 1 : 2 stronger
18 42 3 : 7 regular
80 240 1 : 3 lighter

(To compare 1:2 with 3:7, rewrite them over a common second term: 1:2 = 3.5 : 7, which carries more concentrate than 3 : 7, hence stronger.)

Figure it Out: Ratios & Proportion

Practice

  1. Circle the true statements of proportion:   (i) 5:8 :: 15:24   (ii) 9:4 :: 36:12   (iii) 8:11 :: 11:8   (iv) 16:6 :: 40:15   (v) 14:21 :: 30:12   (vi) 27:9 :: 12:4
  2. Give three ratios proportional to 5 : 7.
  3. Fill in the missing numbers so each ratio is proportional to 16 : 28:   4 : \underline{\ \ }, \quad 8 : \underline{\ \ }, \quad 18 : \underline{\ \ }, \quad 12 : \underline{\ \ }.
  1. Use cross multiplication (a:b :: c:d \iff ad = bc).
      1. 5 \times 24 = 120 = 8 \times 15. True.
      1. 9 \times 12 = 108, 4 \times 36 = 144. False.
      1. 8 \times 8 = 64, 11 \times 11 = 121. False.
      1. 16 \times 15 = 240 = 6 \times 40. True.
      1. 14 \times 12 = 168, 21 \times 30 = 630. False.
      1. 27 \times 4 = 108 = 9 \times 12. True.
    So (i), (iv) and (vi) are true.
  2. Many answers are possible; multiply both terms of 5:7 by the same factor, e.g. 10:14, 15:21, 50:70.
  3. 16 : 28 = 4 : 7 in simplest form, so each proportional ratio is 4:7 scaled.
    • 4 : \mathbf{7} (this is the simplest form).
    • 8 : \mathbf{14} (since 8 = 4 \times 2, second term = 7 \times 2).
    • 18 : \underline{\ \ }, needs 18 = 4 \times \tfrac{18}{4}, so the second term is 7 \times \tfrac{18}{4} = \tfrac{63}{2}, not a whole number; no whole-number partner makes 18 : \underline{\ \ } equal to 4:7.
    • 12 : \mathbf{21} (since 12 = 4 \times 3, second term = 7 \times 3).