Decimal Division

4.3 Decimal Division

Sharing Ribbon

We know that the fraction \(\frac{1}{2}\) can be represented as a decimal 0.5. So, each girl will get 14 m and an additional 0.5 m of ribbon. Hence, the length of ribbon each will get is \(14 + 0.5 = 14.5\) m.

Now, what if the ribbon was shared between four friends instead of 2? So, each will get \(29 \div 4\) m, that is \(\frac{29}{4}\) m.

Now, the denominator of the fraction is 4. To convert a fraction to a decimal, it helps if the denominator is of the form \(\frac{1}{10}\), \(\frac{1}{100}\), 1000 and so on. Can we find a fraction equivalent to \(\frac{29}{4}\) with such a denominator? — Yes!

Is 4 a factor of 100? Yes (4 × 25 = 100). So we can get an equivalent fraction to \(\frac{29}{4}\) with denominator 100 by multiplying the numerator and denominator by 25.

\(\frac{29}{4} = \frac{29 \times 25}{4 \times 25} = \frac{725}{100} = 7.25\)

So each of the 4 friends will get 7.25 m of ribbon.

Division Using Place Value

We have seen how to divide two counting numbers to get a decimal quotient. We first represented the division as a fraction. Then we found an equivalent fraction with denominator of the form 1, 10, 100, and so on, to convert it to a decimal.

Now, let us look at the division using place value procedure to calculate the decimal quotient.

Suppose we want to write the quotient \(\frac{13}{8}\) as a decimal. We can convert this fraction to an equivalent fraction with a denominator such as 1, 10, 100, 1000, etc.

It is not possible. So, we need a more general method to divide any two counting numbers. Let us see how we can use division using place value for this.

Example 5: Long Division of 237 ÷ 8

Find the value of \(237 \div 8\). This extends the division using place value to find quotients with decimals in them. Two hundreds and thirty-seven can be regrouped as tenths, tens and can be regrouped as hundredths and so on.

Steps explained:

• 20 Tens ÷ 8 = 2 Tens, 4 Tens remain.
• 4 Tens + 3 Tens (from regrouping) → 23 Tens; 23 ÷ 8 = 2 Tens, 7 Tens remain.
• 7 Tens = 70 Ones; with 7 original Ones → 77 Ones. 77 ÷ 8 = 9 Ones, 5 Ones remain.
• 5 Ones can't be divided by 8 in the quotient. When we need to regroup into 50 Tenths, we place a decimal point in the quotient.
• 50 Tenths ÷ 8 = 6 Tenths, 2 Tenths remain.
• 2 Tenths cannot be divided into 8 equal parts, so we need to regroup them as 20 Hundredths. 20 Hundredths ÷ 8 = 2 Hundredths, 4 Hundredths remain.
• 4 Hundredths → 40 Thousandths → 5 Thousandths, 0 remaining. Done.

237 ÷ 8 = 29.625

Why Do We Put a Decimal Point in the Quotient?

When we regroup 5 Ones into 50 Tenths, we place a decimal point in the quotient. This is because the quotient now contains a fractional part — every digit written after the point is in the Tenths, Hundredths, Thousandths place.

Example 6: Sridharacharya's Problem (8th century)

The great Indian mathematician Sridharacharya, in his book Patiganita, sets: \(6\frac{1}{4}\) is divided by \(2\frac{1}{2}\), and \(60\frac{1}{4}\) is divided by \(5\frac{3}{4}\). Tell the quotients separately. Can you try to solve it by converting the fractions into decimals?

Cyclic Numbers

What are the products? What do you notice?

You get the same number back, but with the digits cycled around! Multiply \(142857\) by 2, 3, 4, 5, 6 and observe.

Are there other such numbers? To find one such number, we can find \(1 \div 17\) in decimal, and use the repeating block.

Dividend, Divisor, and Quotient

When we divide two counting numbers, the quotient is always less than the dividend. For example, \(128 \div 4 = 32\), and \(32\) (quotient) < \(128\) (dividend).

But what happens when we divide 128 by 0.4?

\(128 \div 0.4 = 320\)

The quotient is greater than the dividend!

Will the quotient always be greater than the dividend when the divisor is a decimal? Try it out with different examples. Describe the relationship between the dividend, divisor, and the quotient. Create a table for capturing this relationship in different situations, like we did for multiplication.

Figure it Out — Section 4.3