NCERT Class 7 Ganita Prakash 2 Chapter 04 Another Peek Beyond the Point PDF Download

Chapter 4: Another Peek Beyond the Point

4.1 A Quick Recap of Decimals

Recall that decimals are the natural extension of the Indian place value system to represent decimal fractions (\( \frac{1}{10}, \frac{1}{100}, \frac{1}{1000} \), 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 with decimals. You will see that the procedures for multiplying and dividing decimals are natural extensions of the procedures for multiplying and dividing counting numbers.

Activity

Question 1: Jonali and Pallabi play a game. Jonali says a fraction and Pallabi gives the equivalent decimal. Write Pallabi's answer in the blank spaces.

Fractions	Decimals
\( \frac{3}{10} \)	0.3...
\( \frac{4}{100} \)	.................
\( \frac{67}{1000} \)	.................
\( \frac{457}{100} \)	.................
\( \frac{71}{100} \)	.................
\( \frac{43}{100} \)	.................
\( \frac{9}{100} \)	.................

Question 2: 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.

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

Activity

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

Fraction	Expanding the Numerator	Sum of one-tenths, one-hundredths, one-thousands,...	Decimals
\( \frac{254}{1000} \)	\( \frac{200}{1000} + \frac{50}{1000} + \frac{4}{1000} = \frac{2}{10} + \frac{5}{100} + \frac{4}{1000} \)	0.2 + 0.05 + 0.004	0.254
\( \frac{847}{10000} \)
\( \frac{173}{100} \)
\( \frac{23}{1000} \)

Math Talk

Can you give a simple rule to divide any number by a number of the form 1 followed by zeroes - 10, 100, 1000, etc.? For example, \( \frac{123}{10} \), \( \frac{24}{100} \) or \( \frac{678}{1000} \)? Look for a pattern in the previous problems.

Here is one such rule. Let us consider the example \( 123 \div 10 \).

Step 1: Write the dividend as it is and place a decimal point at the end.

123.

Step 2: Count the number of zeroes in the divisor.

\( 10 \rightarrow \) 1 zero

Step 3: Move the decimal point from Step 1 left by the same number of places as the count from Step 2. Add zeroes in front if needed.

12.3

Examples

\( 24 \div 100 = 0.24 \) \( \qquad \) \( 678 \div 1000 = 0.678 \)

\( 12 \div 1000 = 0.012 \) \( \qquad \) \( 12345 \div 1000 = 12.345 \)

Teacher's Note

When dividing by 10, 100, 1000, you are just moving the decimal point. Count the zeros in the divisor carefully: 10 has 1 zero (move 1 place), 100 has 2 zeros (move 2 places), and so on. If you run out of digits, add zeros to the left.

4.2 Decimal Multiplication

Example 1: Arshad goes to a stationery shop and purchases 5 pens. If one pen costs Rs. 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, 5 times. That is \( 9.5 \times 5 = 9.5 + 9.5 + 9.5 + 9.5 + 9.5 = 47.5 \).

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

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

The cost of 5 pens = \( \frac{5}{1} \times \frac{95}{10} \).

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

\[ \frac{5}{1} \times \frac{95}{10} = \frac{5 \times 95}{1 \times 10} = \frac{475}{10} = 47.5 \]

The cost of 5 pens is Rs. 47.5.

Example 2: A car travels 12.5 km per litre of petrol. What is the distance covered with 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.

Teacher's Note

To multiply decimals, convert them to fractions with denominators of 10, 100, 1000, etc., multiply the numerators and denominators separately, then convert back to a decimal. The final decimal point position equals the sum of decimal places in both original numbers: \( 12.5 \times 7.5 \) has 1 + 1 = 2 decimal places, giving 93.75.

Key Points

• Decimals extend the place value system to represent fractions with denominators of 10, 100, 1000, and so on.
• To divide a number by 10, 100, or 1000, move the decimal point left by the number of zeros in the divisor.
• To multiply decimals, convert them to fractions, multiply numerators and denominators, then convert back to decimal form.
• The number of decimal places in the product equals the sum of decimal places in the two numbers being multiplied.

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