2.2 Percentages, Ratios, Rates & Proportional Reasoning

Percentages, ratios, and rates are fundamental mathematical tools used to compare quantities and understand relationships between numbers. In the quantitative reasoning section of the Nigeria Police Exam, you will encounter numerous problems requiring you to manipulate these concepts. This section will provide a comprehensive and detailed breakdown of these topics, ensuring you are fully prepared to tackle them with confidence and precision.

Key Percentage and Ratio Formulas

• Percentage Value: $\text{Value} = \left(\frac{\text{Percentage}}{100}\right) \times \text{Total}$
• Percentage Change: $\text{Percentage Change} = \left(\frac{\text{Difference}}{\text{Original Value}}\right) \times 100$
• Ratio Representation: Expressed as $a:b$ or $\frac{a}{b}$, representing proportional parts of a whole.
• Proportionality: Direct proportion maintains a constant ratio ($\frac{y}{x} = k$), while inverse proportion maintains a constant product ($x \cdot y = k$).

Understanding Percentages

The word 'percent' literally means 'for every 100' or 'out of 100'. Therefore, a percentage is a way of expressing a number as a fraction of 100. It is denoted using the percent sign (%). For example, 45% is equivalent to the fraction 45/100, or the decimal 0.45. Converting between percentages, fractions, and decimals is a critical skill. To convert a percentage to a decimal, divide by 100 (or move the decimal point two places to the left). To convert a decimal to a percentage, multiply by 100. To convert a fraction to a percentage, multiply the fraction by 100 and add the % sign.

Calculating a percentage of a given quantity is a common task. To find x% of y, you can use the formula: $(x / 100) \times y$. For instance, to find 15% of 200, you would calculate $(15 / 100) \times 200 = 0.15 \times 200 = 30$. Alternatively, you can use fractional equivalents for common percentages to speed up calculations. Knowing that 25% is 1/4, 50% is 1/2, and 75% is 3/4 can save valuable time during the exam.

Percentage Increase and Decrease

Problems involving percentage increase (such as profit, markup, or growth) and percentage decrease (such as loss, discount, or depreciation) are very common. The formula for calculating the percentage change is:

Percentage Change = (Difference / Original Value) \times 100

Let us explore a detailed example of percentage increase. Suppose a shopkeeper buys a bag of rice for ₦25,000 and sells it for ₦30,000. To find the percentage profit, first find the difference (profit): ₦30,000 - ₦25,000 = ₦5,000. Next, divide this difference by the original cost price and multiply by 100: (5,000 / 25,000) \times 100. Simplifying the fraction gives 1/5. Thus, (1/5) \times 100 = 20%. The percentage profit is 20%.

For a percentage decrease example, consider a pair of shoes originally priced at ₦12,000, now on sale for ₦9,000. The difference (discount) is ₦12,000 - ₦9,000 = ₦3,000. The percentage discount is (3,000 / 12,000) \times 100. Simplifying the fraction gives 1/4. Therefore, (1/4) \times 100 = 25%. The shoes are discounted by 25%.

Ratios

A ratio is a mathematical comparison of two or more quantities of the same units. It indicates how many times one number contains another. Ratios can be expressed in several ways: using a colon (e.g., 3:2), using the word 'to' (e.g., 3 to 2), or as a fraction (e.g., 3/2). It is crucial to remember that order matters in a ratio. The ratio of boys to girls in a class is not the same as the ratio of girls to boys.

Ratios should always be simplified to their lowest terms, just like fractions. For example, a ratio of 20:30 simplifies to 2:3 by dividing both sides by their greatest common factor, which is 10. When solving problems involving ratios, it is often helpful to introduce a constant multiplier, say 'x'. If the ratio of two numbers is 4:5, you can represent the numbers as 4x and 5x. If you know their sum is 90, you can set up the equation: 4x + 5x = 90. This simplifies to 9x = 90, which means x = 10. The numbers are therefore 40 and 50.

Rates

While a ratio compares quantities of the same kind, a rate compares quantities of different kinds, often involving time. Common examples include speed (distance per unit of time, like km/h), wage (money per unit of time, like ₦/hour), and fuel consumption (distance per unit of volume, like km/liter). A unit rate is a rate where the second quantity is one unit, such as 60 kilometers per 1 hour.

To solve rate problems, ensure units are consistent. If a car travels 150 kilometers in 3 hours, its average rate of speed is 150 km / 3 hours = 50 km/h. If you are asked how far the car will travel in 5 hours at the same rate, you multiply the unit rate by the new time: 50 km/h \times 5 hours = 250 kilometers.

Proportional Reasoning

A proportion is an equation stating that two ratios or rates are equal. Proportions are typically written in the format a/b = c/d or a:b :: c:d. Proportional reasoning is the ability to use proportions to solve for an unknown value. The most common method for solving a proportion is cross-multiplication. If a/b = c/d, then a \times d = b \times c.

Let us examine a detailed proportional reasoning problem. If 5 workers can build a wall in 12 days, how long will it take 8 workers to build the same wall, assuming they work at the same rate? This is a problem of inverse proportion because as the number of workers increases, the time required decreases. In inverse proportion, the product of the two variables is constant. Therefore, Workers1 \times Time1 = Workers2 \times Time2. We have 5 \times 12 = 8 \times Time2. This gives 60 = 8 \times Time2. Solving for Time2, we get 60 / 8 = 7.5 days. Understanding the difference between direct proportion (where variables increase or decrease together) and inverse proportion is vital for success.

By mastering percentages, ratios, rates, and proportional reasoning, you equip yourself with powerful analytical tools. Practice converting between formats, setting up ratio equations, and identifying proportional relationships in word problems to maximize your score in the quantitative reasoning section.