2.3 Fractions, Decimals & Averages (Mean, Median, Mode)

Fractions and decimals are essential components of quantitative reasoning. In the Nigeria Police Exam, you will frequently encounter questions that require you to convert between fractions, decimals, and percentages, as well as perform complex operations with them. Furthermore, understanding the various types of averages is critical for interpreting data and solving statistical word problems. This section is designed to be comprehensive, exceeding standard explanations to ensure deep conceptual understanding. Let us explore these topics with rigor and detail.

Summary of Fraction Operations and Averages

• Adding/Subtracting Fractions: Find the Least Common Denominator (LCD), convert numerators, and combine over the common denominator.
• Multiplying Fractions: Multiply numerators together and denominators together directly: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$.
• Dividing Fractions: Multiply by the reciprocal of the divisor: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$.
• Statistical Averages: Mean is sum divided by count; Median is middle value in an ordered set; Mode is the most frequent value.

Demystifying Fractions

A fraction represents a part of a whole or a proportion. It consists of two numbers separated by a line: the numerator (the top number) represents the number of parts we have, and the denominator (the bottom number) represents the total number of equal parts that make up a whole. Fractions can be categorized into three main types. Proper fractions have a numerator smaller than the denominator, representing a value less than 1 (e.g., 3/4). Improper fractions have a numerator equal to or greater than the denominator, representing a value equal to or greater than 1 (e.g., 5/3). Mixed numbers consist of a whole number combined with a proper fraction (e.g., 2 1/2).

Converting between improper fractions and mixed numbers is a fundamental skill. To convert an improper fraction like 17/5 to a mixed number, divide the numerator by the denominator. 17 divided by 5 is 3 with a remainder of 2. The quotient (3) becomes the whole number, the remainder (2) becomes the new numerator, and the denominator remains the same. Thus, 17/5 equals 3 2/5. To convert a mixed number like 4 3/7 to an improper fraction, multiply the whole number by the denominator and add the numerator. $(4 \times 7) + 3 = 28 + 3 = 31$. Place this result over the original denominator to get 31/7.

Operations with Fractions

Adding and subtracting fractions require a common denominator. If the denominators are the same, simply add or subtract the numerators and keep the denominator. If they are different, you must find the Least Common Denominator (LCD), which is the Least Common Multiple (LCM) of the denominators. Let us add 2/3 and 1/4. The LCD of 3 and 4 is 12. Convert each fraction: 2/3 becomes 8/12 (multiply top and bottom by 4), and 1/4 becomes 3/12 (multiply top and bottom by 3). Now add: 8/12 + 3/12 = 11/12.

Multiplying fractions is straightforward: multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. For example, 3/5 \times 2/7 = (3 \times 2) / (5 \times 7) = 6/35. Always simplify the final answer if possible. Dividing fractions involves multiplying the first fraction by the reciprocal of the second fraction (flip the numerator and denominator of the second fraction). To calculate 3/4 \div 5/8, change it to multiplication: 3/4 \times 8/5. The result is (3 \times 8) / (4 \times 5) = 24/20. Simplify by dividing the numerator and denominator by their greatest common factor, 4, resulting in 6/5 or 1 1/5.

Decimals in Detail

Decimals are an alternative way to express fractions based on the base-10 number system. Each position to the right of the decimal point represents a fraction with a denominator that is a power of 10 (tenths, hundredths, thousandths, etc.). For instance, 0.75 represents 75/100, which simplifies to 3/4.

Adding and subtracting decimals is simple: vertically align the decimal points before performing the operation. This ensures that you are adding or subtracting tenths from tenths, hundredths from hundredths, and so forth. If necessary, pad numbers with trailing zeros to maintain alignment. For example, to add 4.2 and 1.35, align them as 4.20 + 1.35 = 5.55.

Multiplying decimals involves ignoring the decimal points initially and multiplying the numbers as if they were whole numbers. After finding the product, count the total number of decimal places in the original numbers. Place the decimal point in the product so that it has the same total number of decimal places. For example, $0.4 \times 0.03$. Multiply 4 by 3 to get 12. There is one decimal place in 0.4 and two in 0.03, totaling three decimal places. Move the decimal point three places to the left in 12 to get 0.012. Dividing decimals usually requires adjusting the divisor to a whole number by moving its decimal point to the right. Move the decimal point in the dividend the same number of places, then perform long division. Ensure you maintain accurate placement throughout the calculation.

Mastery of Averages: Mean, Median, and Mode

Statistical averages are critical for summarizing data sets. The term 'average' usually refers to the arithmetic mean, but median and mode are equally important concepts that test different aspects of data interpretation.

The Mean is the most common measure of central tendency. It is calculated by adding all the values in a data set and dividing the sum by the total number of values. For example, find the mean of 12, 15, 18, 22, and 28. First, find the sum: $12 + 15 + 18 + 22 + 28 = 95$. Since there are 5 values, divide the sum by 5: $95 \div 5 = 19$. The mean is 19. Be prepared for problems that require you to find a missing value when the mean is known. If the mean of four numbers is 10, their sum must be 40 ($4 \times 10$). If three of the numbers are 8, 12, and 15, their sum is 35. The missing number is $40 - 35 = 5$.

The Median is the middle value in a data set when the values are arranged in ascending or descending numerical order. If the data set has an odd number of values, the median is the exact middle number. For the set 4, 7, 9, 13, 20, the median is 9. If the data set has an even number of values, the median is the arithmetic mean of the two middle numbers. For the set 3, 6, 8, 10, 14, 18, the two middle numbers are 8 and 10. The median is $(8 + 10) / 2 = 9$. The median is particularly useful because, unlike the mean, it is not significantly affected by extreme values (outliers).

The Mode is the value that appears most frequently in a data set. A data set may have one mode (unimodal), multiple modes (bimodal or multimodal), or no mode if all values appear with the same frequency. In the set 5, 8, 9, 8, 12, 14, 8, the mode is 8 because it appears three times, which is more frequent than any other number. The mode is the only measure of central tendency that can be used with categorical (non-numerical) data.

By deeply understanding fractions, decimals, and the distinct characteristics of mean, median, and mode, you will be well-equipped to handle a vast array of quantitative problems on the exam.