5.2 Ratios, Proportions & Percentages

Ratios, Proportions, and Percentages Overview

Ratios, proportions, and percentages represent a significant portion of the arithmetic problems on the Saudi Arabia Qiyas General Aptitude Test (GAT). These concepts test your ability to make relative comparisons, scale values, and calculate changes in quantities. GAT questions are structured to reward candidates who understand the relationships between ratios and parts, direct and inverse proportions, and successive percentage updates.

Understanding Ratios & Combined Ratios

A ratio is a comparison of two or more quantities by division, written as a:b or a/b. GAT word problems require distinguishing between part-to-part and part-to-whole relationships. For example, if the ratio of boys to girls in a class is 3:5, the part-to-part ratio is 3 boys for every 5 girls. The part-to-whole relationships are 3/8 for boys and 5/8 for girls, where the denominator is the sum of the ratio parts (3 + 5 = 8). Understanding this difference is critical because GAT questions will often provide a part-to-part ratio but ask for a fraction or percentage of the entire group. A common GAT problem involves combining ratios A:B and B:C to find the combined ratio A:B:C. Example: The ratio of Ahmed's savings to Khaled's savings is 2:3, and the ratio of Khaled's savings to Fahad's savings is 4:5. What is the ratio of Ahmed's savings to Fahad's savings? To combine these, find a common multiplier for the shared variable (Khaled, who is represented by 3 in the first ratio and 4 in the second). The least common multiple (LCM) of 3 and 4 is 12. Multiply the first ratio (2:3) by 4 to get 8:12. Multiply the second ratio (4:5) by 3 to get 12:15. The combined ratio Ahmed:Khaled:Fahad is 8:12:15. Thus, the ratio of Ahmed to Fahad is 8:15. If the question asked for Fahad's share as a fraction of the total savings, it would be 15 out of 35 (8 + 12 + 15 = 35), which simplifies to 3/7.

To partition a total:

1. Sum the ratio parts.
2. Divide the total by this sum to find one part's value.
3. Multiply this value by each ratio term. Example: A father distributes 24,000 SAR among his three sons in the ratio 2:3:5. How much does the second son receive?
4. Sum of parts: 2 + 3 + 5 = 10 parts.
5. Value of one part: 24,000 SAR / 10 = 2,400 SAR.
6. The second son receives 3 parts: 3 * 2,400 SAR = 7,200 SAR.

Proportions: Direct and Inverse

A proportion equates two ratios. GAT tests two proportional relationships:

1. Direct Proportion: As one quantity increases, the other increases. The ratio y/x remains constant (y1/x1 = y2/x2). Solve by cross-multiplication. Example: If 5 books cost 75 SAR, 8 books cost x. So, 5/75 = 8/x, which yields 5x = 600, or x = 120 SAR.
2. Inverse Proportion: As one quantity increases, the other decreases. The product x * y remains constant (x1y1 = x2y2). Solve by horizontal multiplication. Example: If 15 workers harvest a field in 8 days, 10 workers take x days. So, 158 = 10x, which yields 120 = 10x, or x = 12 days.

Percentage Basics and Conversions

Percentages are fractions with a denominator of 100. Succeeding in GAT requires memorizing common fraction-to-percentage conversions to bypass calculations:

• 1/2 = 50%
• 1/3 = 33.3%
• 1/4 = 25%
• 1/5 = 20%
• 1/8 = 12.5%
• 1/10 = 10%

Percentage change measures the relative difference between original and new values: Percentage Change = (Change / Original Value) * 100 Always use the original value as the denominator. Example: If a price increases from 80 SAR to 100 SAR, the change is 20 SAR. The percentage increase is (20 / 80) * 100 = 25%. Using 100 as the base is a common error that incorrectly yields 20%.

Successive Percentage Changes

Successive percentage changes, like consecutive discounts, are never additive. Example: A 20% discount followed by a 10% discount on an item of 100 SAR:

1. The first 20% discount reduces the price to 80 SAR (100 * 0.8).
2. The second 10% discount is applied to the 80 SAR: 10% of 80 is 8 SAR.
3. The final price is 80 - 8 = 72 SAR. The total discount is 100 - 72 = 28 SAR (28% overall discount, not 30%). Mathematically, the multiplier is (1 - 0.20) * (1 - 0.10) = 0.8 * 0.9 = 0.72.

To find the percentage of a percentage, multiply the decimals or fractions. Example: What is 40% of 30% of 500? Convert the percentages to decimals: 0.40 * 0.30 * 500 = 0.12 * 500 = 60.

Common Qiyas Ratio and Percentage Traps

• The Reversal Trap: If a value decreases by 20%, it does not return to its original value with a 20% increase. For instance, if 100 decreases by 20%, it becomes 80. To return 80 to 100, you must increase it by 20, which is 20/80 = 25%.
• Direct Percentage Addition: Never add successive percentages directly. Always multiply their corresponding multipliers.
• Linear vs. Area Ratios: If the side length of a square increases by 50%, the area does not increase by 50%. The area scales by the square of the linear scale factor. If side goes from 1 to 1.5, the area goes from 1^2 to 1.5^2 = 2.25, representing a 125% increase. For volume, the scale factor is cubed.