Recap and Decimal Multiplication

4.1 A Quick Recap of Decimals

Recall that decimals^? are the natural extension of the Indian place-value system to represent decimal fractions \(\left(\frac{1}{10}, \frac{1}{100}, \frac{1}{1000}\right.\) and so on) and their sums.

For example, 27.53 refers to a quantity that has: 2 Tens, 7 Units (Ones), 5 Tenths, 3 Hundredths.

We have already learned how to multiply and divide fractions. In this chapter, we will learn how to perform these operations on decimals. You will see that the procedures for multiplying and dividing decimals are simply an extension of the procedures we already know for multiplying and dividing counting numbers.

Jonali & Pallabi: Market Shopping

Jonali and Pallabi play a game. Jonali says a fraction and Pallabi gives the equivalent decimal. Pallabi's answers fill in the blank spaces.

Jonali goes to the market to buy spices. She purchases: 50 g of Cinnamon, 100 g of Cumin seeds, 25 g of Cardamom, and 250 g of Pepper. Express each of the quantities in kilograms by writing them in terms of fractions as well as decimals. (1 kg = 1000 g.)

The fractions Jonali gave Pallabi have denominators 10, 100, 1000, and so on.

Expanding Fractions as Sums: From Fraction to Decimal

Write the following fractions as a sum of fractions and also as decimals.

4.2 Decimal Multiplication

Example 1: Arshad's Pens

Arshad goes to a stationary shop and purchases 5 pens. If one pen costs ₹9.5 (9 rupees and 50 paisa), how much should he pay the shopkeeper?

What operation must we use here? We have to multiply 9.5 by 5, which is the same as adding 9.5 five times.

\(9.5 \times 5 = 9.5 + 9.5 + 9.5 + 9.5 + 9.5 = 47.5\)

We can also directly multiply the numbers by converting them into fractions.

\(9.5 = \frac{95}{10}\) and 5 is \(\frac{5}{1}\) as a fraction. So:

\(\frac{5}{1} \times \frac{95}{10} = \frac{5 \times 95}{1 \times 10} = \frac{475}{10} = 47.5\)

The cost of 5 pens is ₹47.50.

Recall that, to find the product of two fractions, we multiply the numerators and multiply the denominators.

Example 2: Car Petrol

A car travels 12.5 km per litre of petrol. What is the distance covered in 7.5 litres of petrol?

We have to multiply 12.5 by 7.5.

The distance covered = \(12.5 \times 7.5 = \frac{125}{10} \times \frac{75}{10} = \frac{125 \times 75}{10 \times 10} = \frac{9375}{100} = 93.75\).

The distance covered is 93.75 km.

Example 3: Ajay's Walk to School

The distance between Ajay's school and his home is 827 m. He walks to school in the morning and then walks back home in the evening, 6 days a week. How much does Ajay walk in a week? Answer in kilometres.

Each way between school and home, Ajay walks 827 metres, i.e., 0.827 km. So, in a day he walks:

\(0.827 \times 2 = \frac{827}{1000} \times 2 = \frac{827 \times 2}{1000} = \frac{1654}{1000} = 1.654\) km

He goes to school 6 days a week. So, in a week he walks:

\(1.654 \times 6 = \frac{1654}{1000} \times 6 = \frac{9924}{1000} = 9.924\) km.

Ajay walks 9.924 km a week.

Example 4: Area of a Rectangle

Find the area of the given rectangle with length 13.3 cm and breadth 5.7 cm.

The area = length × breadth = \(5.7 \times 13.3 = \frac{57}{10} \times \frac{133}{10} = \frac{7581}{100} = 75.81\) sq cm.

Rule: Counting Decimal Places in Products

Observe the number of digits after the decimal point in the multiplier, the multiplicand and the product. Also note the number of zeros in the denominator.

Suppose we know that \(596 \times 248 = 147808\). Can you immediately write down the product of \(5.96 \times 24.8\)?

Example: To evaluate \(5.96 \times 24.8\):

\(596 \times 248 = 147808\) → 2 + 1 = 3 decimal places → 5.96 × 24.8 = 147.808