Proportional Reasoning-1 · Lesson 3 of 8
Problem Solving with Proportional Reasoning
“Use equivalent ratios and scale factors in recipes, fair comparisons, changing ages, missing-term problems and visual investigations.”
• Choose the quantities and order in a ratio problem. • Scale recipes without changing their proportions. • Explain why equal age increases do not preserve an age ratio. • Compare mixture strengths and complete missing ratio terms. • Use ratios in drawings and repeating patterns.
Two cups can hold different amounts of lemonade yet taste equally sweet. Two walls can use different amounts of cement without that difference alone showing that one is worse. In both situations, comparing just one number misses the relationship between two quantities. Ratios let you make a fairer comparison. The first step is to decide what must stay the same, then check how the paired quantities change together.
You will use simplification and common scale factors throughout this lesson. Before doing arithmetic, say what each number measures and why a proportional relationship is appropriate for the situation.
7.4 Problem Solving with Proportional Reasoning
Problem
Are 3 : 4 and 72 : 96 proportional?
For a recipe, keeping the ingredient proportions constant means scaling the whole batch consistently. Assume the glasses have the same capacity and each spoon measures the same amount of sugar. Otherwise the counts alone would not describe equal quantities.
Kesang uses 10 spoons of sugar for six glasses. Her father asks for 18 more glasses. The question concerns the new batch, so compare six glasses with 18 additional glasses. Do not mix up the added amount and the combined total.
Problem
Six glasses use 10 spoons of sugar. How many spoons are needed for 18 additional glasses?
Now compare a 60 ft wall using three bags of cement with a 40 ft wall using two bags. Length : bags simplifies to 20 : 1 in both cases. The shorter wall does not use less cement per foot.
This model assumes matching wall height, thickness, materials and construction conditions. Equal length-to-cement ratios alone do not prove equal real-world strength; they establish equal cement use relative to wall length under the comparison’s assumptions.
Problem
One wall is 60 ft long and uses 3 bags; another is 40 ft long and uses 2 bags. Compare their ratios.
Not every pair of changing numbers stays proportional. When Neelima is three, her mother is 30. Nine years later their ages become 12 and 39. The same addition preserves their age difference of 27 years, but not their age ratio.
Originally 3 : 30 simplifies to 1 : 10. Later 12 : 39 simplifies to 4 : 13. A constant difference and a constant ratio are different relationships. Adding the same amount to both terms generally changes a ratio; multiplying both by the same factor preserves it.
Problem
Neelima is 3 and her mother is 30. Find their age ratio when Neelima is 12.
Problem
Complete __ : 42, 6 : __ and 2 : __ so each is proportional to 14 : 21.
Filter Coffee!
Regular filter coffee uses 15 mL of decoction with 35 mL of milk. Its decoction-to-milk ratio is 15 : 35 = 3 : 7. Stronger coffee uses more decoction relative to milk, not merely a greater total amount of decoction.
The 20 mL–30 mL mixture has ratio 2 : 3 and is stronger than the regular mixture. The 10 mL–40 mL mixture has ratio 1 : 4 and is lighter. For these three 50 mL cups, you can compare the decoction amounts directly because the total volumes are equal.
For different batch sizes, compare at a common amount of milk or at a common total volume. This avoids a trap: 150 mL of decoction may make a lighter mixture than 24 mL when the milk amounts differ enough.
Problem
Classify 300 mL decoction with 600 mL milk, and 24 mL decoction with 56 mL milk, compared with regular 15 : 35 coffee.
| Decoction (mL) | Milk (mL) | Simplest ratio | Compared with regular 3 : 7 |
|---|---|---|---|
| 300 | 600 | 1 : 2 | Stronger |
| 150 | 500 | 3 : 10 | Lighter |
| 200 | 400 | 1 : 2 | Stronger |
| 24 | 56 | 3 : 7 | Regular |
| 100 | 300 | 1 : 3 | Lighter |
Figure it Out
A repeating wall pattern can be analysed by choosing one complete repeating unit. In the first supplied pattern, a unit contains nine grey and six coloured bricks, giving 3 : 2. In the second, 16 grey and 12 coloured bricks give 4 : 3.
Repeating a complete unit multiplies both counts equally. Counting an arbitrary cropped section can cut off part of the pattern and give a misleading ratio. Explain the choice of repeat unit before simplifying its counts.
Quiz
Six glasses use 10 sugar spoons. How many additional spoons make 18 additional glasses equally sweet?
What stays constant when both people grow nine years older?
Which ratio completes 14 : 21 :: 6 : __?
Which batch has regular strength compared with decoction : milk = 3 : 7?
Why count a complete repeat unit in a brick pattern?
Practice Problems
- Are 21 : 6 and 35 : 10 proportional?
- Complete 18 : 24 :: 20 : x.
- A recipe uses 8 spoons for 12 equal glasses. How many spoons are needed for 30 equal glasses?
- Compare 150 mL decoction with 500 mL milk against regular 3 : 7 coffee.
- A child is 4 and a parent is 28. Find their ratio now and eight years later. Explain the change.
1. Their simplest forms are 7 : 2 and 7 : 2. 2. Yes; the comparison is unchanged.
1. 18 : 24 simplifies to 3 : 4. The first term 3 becomes 20 by ×20/3. 2. The second becomes 4 × 20/3 = 80/3, so x = 26⅔.
1. Scale factor = 30/12 = 5/2. 2. Sugar = 8 × 5/2 = 20 spoons.
1. The batch ratio is 3 : 10. 2. For the same three decoction parts, it has ten milk parts rather than seven, so it is lighter.
1. Now the ratio is 4 : 28 = 1 : 7. 2. Later the ages are 12 and 36, giving 1 : 3. 3. The difference remains 24 years, but adding eight to both ages does not preserve their ratio.
Key Takeaways
• Choose consistent quantity labels before comparing numbers. • A common scale factor preserves a recipe’s proportions. • More of one ingredient does not by itself mean a stronger mixture. • Equal additions preserve a difference but generally change a ratio. • Equivalent ratios may require fractional scale factors and fractional terms. • Use complete repeating units and corresponding measurements in visual comparisons.