Fractions and decimals are two different ways of representing parts of a whole. A fraction is written in the form p/q where q is not equal to 0, and it represents a portion of a quantity. For example, if a pizza is cut into 6 equal slices and you eat 2 of them, you have eaten 2/6 of the pizza, which simplifies to 1/3. Decimals are another way to write fractions, such as 0.5 for 1/2, 0.75 for 3/4 and 0.125 for 1/8. Both fractions and decimals appear constantly in everyday life: in measurements of length, weight, capacity, time, money, and in recipes, shopping bills and sports statistics.
In Class 7 we learn how to multiply and divide fractions, convert between fractions and decimals, perform the four operations on decimals, and solve real-life problems involving these numbers. A solid understanding of fractions and decimals is essential because nearly every chapter ahead, from comparing quantities to perimeter and area, uses these skills. The key ideas are equivalence, simplification, and careful placement of the decimal point.
To multiply a fraction by a whole number, multiply the numerator by the whole number and keep the denominator unchanged: (a/b) x c = (a x c) / b For example, 3 x (2/5) = 6/5. If possible, reduce the answer to its simplest form.
To multiply two fractions, multiply the numerators together and the denominators together: (a/b) x (c/d) = (a x c) / (b x d) For example, (2/3) x (4/5) = 8/15. Before multiplying, you may cancel common factors between any numerator and any denominator to make the calculation simpler. This is called cross-cancelling.
The same rule extends to more than two fractions. Multiply all numerators and all denominators, cancelling common factors wherever possible. For example, (1/2) x (2/3) x (3/4) = 6/24 = 1/4.
Finding a fraction of a quantity means multiplying the fraction by the quantity. For example, to find 3/4 of 20, we compute (3/4) x 20 = 60/4 = 15.
The reciprocal of a non-zero fraction a/b is b/a. The product of a fraction and its reciprocal is always 1. For example, the reciprocal of 3/4 is 4/3 and (3/4) x (4/3) = 1. The reciprocal of a whole number n is 1/n.
To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole number: (a/b) / c = (a/b) x (1/c) = a / (b x c) For example, (3/4) / 2 = (3/4) x (1/2) = 3/8.
To divide by a fraction, multiply by its reciprocal: (a/b) / (c/d) = (a/b) x (d/c) = (a x d) / (b x c) For example, (5/6) / (2/3) = (5/6) x (3/2) = 15/12 = 5/4.
To divide a whole number by a fraction, multiply the whole number by the reciprocal of the fraction: n / (a/b) = n x (b/a) For example, 4 / (2/3) = 4 x (3/2) = 12/2 = 6.
A decimal number has a whole part and a fractional part separated by a decimal point. Each digit to the right of the decimal point represents tenths, hundredths, thousandths and so on. For example, in 12.345, the digit 3 is in the tenths place, 4 in the hundredths place and 5 in the thousandths place. A decimal can always be converted into a fraction: 0.75 = 75/100 = 3/4.
To multiply two decimals, first multiply them as whole numbers, ignoring the decimal points. Then place the decimal point in the product so that the number of decimal places in the product equals the total number of decimal places in both the factors combined. For example, 2.5 x 1.3: 25 x 13 = 325. The total decimal places are 1 + 1 = 2, so 2.5 x 1.3 = 3.25. When multiplying a decimal by 10, 100 or 1000, shift the decimal point to the right by 1, 2 or 3 places respectively.
To divide a decimal by a whole number, divide as usual and place the decimal point in the quotient directly above the decimal point in the dividend. For example, 7.5 / 3 = 2.5. To divide a decimal by a decimal, shift the decimal point in both the divisor and the dividend to the right by the same number of places so that the divisor becomes a whole number, then divide. For example, 4.8 / 1.2 = 48 / 12 = 4. To divide by 10, 100 or 1000, shift the decimal point to the left by 1, 2 or 3 places respectively.
To add or subtract decimals, write the numbers one below the other such that the decimal points are aligned, filling missing places with zeros, and then add or subtract as with whole numbers. For example, 3.5 + 2.75 is written as 3.50 + 2.75 = 6.25.
A fraction can be converted to a decimal by dividing the numerator by the denominator. For example, 1/4 = 0.25, 3/8 = 0.375. Some fractions give a terminating decimal (like 1/2 = 0.5), while others give a repeating decimal (like 1/3 = 0.333...). A decimal can be converted to a fraction by writing it as a fraction with denominator 10, 100, 1000 and then simplifying. For example, 0.6 = 6/10 = 3/5 and 0.625 = 625/1000 = 5/8.
Word problems about fractions and decimals appear in measurements, money and time. If a recipe needs 3/4 litre of milk and you are making half the recipe, you need (1/2) x (3/4) = 3/8 litre. If a piece of cloth is 12.5 m long and is cut into 5 equal pieces, each piece is 12.5 / 5 = 2.5 m. If you travel 2/5 of a 60 km journey, you have covered (2/5) x 60 = 24 km. Always read the problem to decide whether you need to multiply (of, fraction of, times) or divide (per, into equal parts).
| Operation | Rule | Example |
|---|---|---|
| Fraction x Whole number | (a/b) x c = (a x c)/b | (2/5) x 3 = 6/5 |
| Fraction x Fraction | (a/b) x (c/d) = (a x c)/(b x d) | (2/3) x (4/5) = 8/15 |
| Fraction / Whole number | (a/b) / c = (a/b) x (1/c) | (3/4) / 2 = 3/8 |
| Fraction / Fraction | (a/b) / (c/d) = (a/b) x (d/c) | (5/6) / (2/3) = 5/4 |
| Whole number / Fraction | n / (a/b) = n x (b/a) | 4 / (2/3) = 6 |
| Reciprocal | Reciprocal of a/b is b/a | Reciprocal of 3/4 is 4/3 |
| Operation | Rule | Example |
|---|---|---|
| Multiply | Count total decimal places of both factors | 2.5 x 1.3 = 3.25 |
| Divide by whole number | Place point in quotient above point in dividend | 7.5 / 3 = 2.5 |
| Divide by decimal | Shift points to make divisor whole | 4.8 / 1.2 = 4 |
| Multiply by 10/100/1000 | Move point right 1/2/3 places | 0.25 x 10 = 2.5 |
| Divide by 10/100/1000 | Move point left 1/2/3 places | 2.5 / 100 = 0.025 |
| Add/Subtract | Align decimal points, add zeros | 3.50 + 2.75 = 6.25 |
Fractions and decimals are powerful tools for representing and computing with parts of a whole. Multiplying fractions is done straight across the numerators and denominators, while division requires multiplying by the reciprocal. Decimal operations demand careful attention to the decimal point, which shifts right when multiplying by powers of ten and left when dividing. With regular practice of conversions between fractions and decimals, and by applying these skills to money, measurement and everyday situations, students build the fluency needed for rational numbers, comparing quantities and mensuration in later chapters.