Comparing quantities means deciding which of two quantities is larger, smaller or equal, and by how much. We compare quantities using ratios, percentages and proportions. A ratio compares two quantities of the same kind and is written as a:b or a/b. A percentage is a ratio with denominator 100; the symbol percent means per hundred. For example, if a student scores 80 out of 100, the score is 80 percent. Percentages, ratios and proportions appear everywhere: in discounts on shopping, interest on savings, profit and loss in business, and data in newspapers.
This chapter teaches us to express one quantity as a percentage of another, to increase and decrease quantities by a percentage, to solve profit and loss problems, to compute simple interest, and to interpret ratios as percentages. These skills are essential for every future topic in commercial mathematics and for understanding economic life. Learning to convert fluently between fractions, decimals, ratios and percentages is the heart of this chapter.
A ratio compares two quantities of the same kind and has no unit. It is written as a:b, meaning a divided by b. For example, if there are 30 boys and 20 girls in a class, the ratio of boys to girls is 30:20, which simplifies to 3:2. A ratio must be expressed in its simplest form by dividing both terms by their highest common factor.
A proportion states that two ratios are equal. If a/b = c/d, then a, b, c and d are in proportion, written as a:b = c:d or a:b :: c:d. The terms a and d are called the extremes and b and c are called the means. In a proportion, the product of the extremes equals the product of the means: a x d = b x c For example, 2:3 = 4:6 because 2 x 6 = 3 x 4 = 12.
The unitary method finds the value of one unit first, then the required value. For example, if 5 pens cost 40 rupees, one pen costs 8 rupees, so 8 pens cost 8 x 8 = 64 rupees.
Percent means per hundred. Thus 25 percent means 25 out of every 100. A percentage can be written as a fraction with denominator 100 or as a decimal: 25 percent = 25/100 = 0.25.
To find a percentage of a quantity, convert the percentage to a fraction or decimal and multiply. For example, 20 percent of 250 = (20/100) x 250 = 50. 12 percent of 400 = 0.12 x 400 = 48.
To find what percentage one quantity is of another, divide the first by the second and multiply by 100. For example, 15 out of 60 is (15/60) x 100 = 25 percent. If a student scores 45 out of 50, the percentage is (45/50) x 100 = 90 percent.
To increase a number by a percentage, find the percentage of the number and add it. To decrease, subtract it. For example, increasing 200 by 10 percent gives 200 + 20 = 220. Decreasing 400 by 25 percent gives 400 - 100 = 300. Alternatively, increasing by p percent means multiplying by (1 + p/100), and decreasing means multiplying by (1 - p/100).
The cost price (CP) is the price at which an article is bought, including overheads. The selling price (SP) is the price at which it is sold. If SP > CP, there is a profit; if SP < CP, there is a loss.
Profit = SP - CP Loss = CP - SP Profit percent = (Profit/CP) x 100 Loss percent = (Loss/CP) x 100 For example, if an article is bought for 400 rupees and sold for 480 rupees, the profit is 80 rupees and the profit percent is (80/400) x 100 = 20 percent. If sold for 360 rupees, the loss is 40 rupees and the loss percent is (40/400) x 100 = 10 percent.
If profit percent and CP are known: SP = CP x (1 + profit percent/100). If loss percent and CP are known: SP = CP x (1 - loss percent/100). To find CP from SP and profit percent: CP = (100 x SP)/(100 + profit percent). To find CP from SP and loss percent: CP = (100 x SP)/(100 - loss percent).
When money is deposited or borrowed, interest is the money paid for its use. The principal (P) is the amount of money deposited or borrowed. The rate (R) is the interest on 100 rupees for one year. The time (T) is the duration in years. The total amount (A) is the sum of the principal and the interest: A = P + Interest
Simple Interest = (P x R x T)/100 For example, if 2000 rupees is deposited at 5 percent per annum for 3 years, the interest is (2000 x 5 x 3)/100 = 300 rupees, and the amount is 2000 + 300 = 2300 rupees.
Amount = Principal + Simple Interest. If the time is given in months, convert to years first by dividing by 12, since the rate is per annum (per year).
Percentages and ratios are used to compare marks, examine population growth, compute discounts during sales, calculate tax, and study rates and speeds. A discount is a reduction given on the marked price; if an article marked 500 rupees is sold at a 10 percent discount, the discount is 50 rupees and the selling price is 450 rupees. Understanding these applications helps students make sense of shop bills, bank statements and news reports.
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50 percent |
| 1/4 | 0.25 | 25 percent |
| 3/4 | 0.75 | 75 percent |
| 1/5 | 0.2 | 20 percent |
| 2/5 | 0.4 | 40 percent |
| 1/8 | 0.125 | 12.5 percent |
| 3/8 | 0.375 | 37.5 percent |
| 1/3 | 0.333... | 33.33 percent |
| Concept | Formula |
|---|---|
| Ratio | a:b simplified to lowest terms |
| Proportion | a x d = b x c |
| Percentage of a quantity | (p/100) x quantity |
| One quantity as percentage of another | (part/whole) x 100 |
| Profit | SP - CP |
| Loss | CP - SP |
| Profit percent | (Profit/CP) x 100 |
| Loss percent | (Loss/CP) x 100 |
| Simple interest | (P x R x T)/100 |
| Amount | P + Simple Interest |
Comparing quantities through ratios, proportions and percentages is one of the most practical parts of mathematics. It lets us express marks, discounts, profits, losses and interest in a uniform, understandable way. The ability to move fluently between fractions, decimals and percentages, and to apply formulas for profit, loss and simple interest, equips students for everyday commercial life and for higher studies in mathematics and economics. Every formula in this chapter is built on the simple idea of a part relative to a whole, expressed per hundred. Mastering this chapter makes financial mathematics intuitive and error-free.