SUMMARY : Proportional Reasoning-1 MCQS -2 | Ganita Prakash | Grade 8

Proportional Reasoning: A Complete Guide & Quiz

Have you ever wondered how chefs scale a recipe up for a large party, or how architects create tiny models of huge buildings? The secret lies in a powerful mathematical concept: proportional reasoning. Understanding ratios and proportions is a fundamental skill that applies everywhere, from the kitchen to complex engineering. This guide will break down the core ideas and help you master them with an interactive quiz!

What is a Ratio?

A ratio is simply a way to compare two quantities. We write it in the form a : b. This notation tells us that for every 'a' units of the first quantity, there are 'b' units of the second quantity. The numbers 'a' and 'b' are called the terms of the ratio.

Example: If the ratio of cats to dogs in a shelter is 2 : 5, it means for every 2 cats, there are 5 dogs.

Key Concepts at a Glance

• Ratios (a : b): A comparison showing that for every 'a' units of one thing, there are 'b' units of another.
• Proportional Ratios (a : b :: c : d): Two ratios are proportional if they represent the same relationship. This happens when their terms change by the same multiplicative factor. The ultimate test is the cross-multiplication rule: ad = bc.
• Dividing a Quantity: To divide a total quantity 'x' into a ratio m : n, you calculate the shares as m × (x / (m + n)) for the first part and n × (x / (m + n)) for the second part.

When are Two Ratios Proportional?

This is the heart of proportional reasoning! Two ratios, say a : b and c : d, are proportional if they express the same relationship. Think of them as equivalent fractions. For instance, the ratio 1 : 2 is proportional to 5 : 10 because you can multiply both terms of the first ratio by 5 to get the second one.

The Golden Rule of Proportions: Cross-Multiplication

The easiest way to check if two ratios are proportional is by using cross-multiplication. For two ratios a : b and c : d to be proportional, the product of the "extremes" (a and d) must equal the product of the "means" (b and c).

So, a : b :: c : d is true if and only if ad = bc.

Example: Is 3 : 4 proportional to 9 : 12?
Let's check: ad = 3 × 12 = 36. And bc = 4 × 9 = 36.
Since 36 = 36, the ratios are proportional!

How to Divide a Quantity in a Given Ratio

Imagine you need to share \$100 between two people in the ratio 2 : 3. It's not an even split. Here's the method:

1. Add the ratio terms: m + n = 2 + 3 = 5. This tells you the total number of "parts" in the whole.
2. Find the value of one part: Divide the total quantity by the sum of the parts: $100 / 5 = $20. So, one part is worth $20.
3. Calculate each share: Multiply the value of one part by each term in the ratio.
   • Person 1's share (2 parts): 2 × $20 = $40
   • Person 2's share (3 parts): 3 × $20 = $60

The shares are $40 and $60. Notice that $40 + $60 = $100, and the ratio 40 : 60 simplifies to 2 : 3.

Test Your Knowledge: Proportional Reasoning Quiz!

Ready to apply what you've learned? Click on an option for each question to see if you're right. The correct answer will glow green, and a wrong one will glow red.

1. What do the 'a' and 'b' in a ratio 'a : b' represent?

• The sum of two quantities
• The terms of the ratio
• The product of two quantities
• The difference between two quantities

2. Two ratios, a : b and c : d, are proportional if...

• a + b = c + d
• a - d = c - b
• ad = bc
• ac = bd

3. Which of the following ratios is proportional to 2 : 5?

• 5 : 2
• 4 : 12
• 8 : 15
• 6 : 15

4. To divide a quantity 'x' in the ratio m : n, what is the first part equal to?

• m × (x / n)
• m × (x / (m + n))
• x × (m / n)
• (m + n) / x

5. Are the ratios 4 : 7 and 12 : 21 proportional?

• Yes, because 4 × 21 = 7 × 12
• No, because the terms are different
• No, because 4 + 17 is not equal to 7 + 14
• Yes, because 4 + 8 = 12

6. A recipe calls for 3 cups of flour for every 2 cups of sugar. What is the ratio of flour to sugar?

• 2 : 3
• 3 : 2
• 3 : 5
• 5 : 3

7. Find the missing number in the proportion 5 : 8 :: ? : 32.

• 16
• 24
• 20
• 40

8. Divide 60 items into two parts in the ratio 1 : 4. What are the two parts?

• 10 and 50
• 15 and 45
• 12 and 48
• 20 and 40

9. The notation a : b :: c : d indicates that...

• The ratios are unequal
• The ratios a : b and c : d are proportional
• The sum of the ratios is zero
• The first ratio is larger than the second

10. If a : b is proportional to c : d, what is 'd' equal to?

• ac / b
• ab / c
• bc / a
• a / bc

11. The ratio of boys to girls in a class is 4 : 5. If there are 36 students in total, how many girls are there?

• 16
• 20
• 24
• 5

12. A map scale is 1 cm : 50 km. If two cities are 4.5 cm apart on the map, what is the actual distance?

• 200 km
• 112.5 km
• 225 km
• 250 km

13. Which statement is FALSE regarding ratios and proportions?

• Proportional ratios simplify to the same base ratio.
• The order of terms in a ratio matters.
• Adding the same number to both terms of a ratio keeps it proportional.
• Proportions can be used to find an unknown quantity.

14. If an investment of $5,000 earns a profit of $200, what profit would an investment of $30,000 earn at the same rate?

• $1,000
• $6,000
• $1,500
• $1,200

15. A painter mixes 2 litres of blue paint with 3 litres of yellow paint to make green. To get the same shade of green, how much yellow paint is needed for 8 litres of blue paint?

• 10 litres
• 12 litres
• 9 litres
• 15 litres