Question 16

Let us explore the quotient. It changes with decimal divisors.

Step 1 — Testing with a small decimal divisor

Let us pick a dividend. Let the dividend be 10. Let us pick a decimal divisor. Let the divisor be 0.5. We will divide the dividend by the divisor. This is $10 \div 0.5$. To divide by a decimal. We change the divisor. We multiply both numbers by 10. So, $10 \div 0.5$ becomes $100 \div 5$. Let us calculate the quotient.

$10 \div 0.5$

$= 100 \div 5$

$\boxed{20}$

Here, the quotient is 20. The dividend was 10. The quotient (20) is greater than the dividend (10).

Step 2 — Testing with a large decimal divisor

Let us use the same dividend. The dividend is 10. Let us pick another decimal divisor. Let the divisor be 2.5. We will divide the dividend by this divisor. This is $10 \div 2.5$. To divide by a decimal. We change the divisor. We multiply both numbers by 10. So, $10 \div 2.5$ becomes $100 \div 25$. Let us calculate the quotient.

$10 \div 2.5$

$= 100 \div 25$

$\boxed{4}$

Here, the quotient is 4. The dividend was 10. The quotient (4) is smaller than the dividend (10).

Step 3 — Answering the first question

In Step 1, the quotient was greater than the dividend. In Step 2, the quotient was smaller than the dividend. So, the quotient is not always greater than the dividend. This happens when the divisor is a decimal. It depends on the value of the decimal divisor.

Step 4 — Describing the relationship

Division finds how many times one number fits into another. We have a dividend, a divisor, and a quotient. The relationship is: Dividend $\div$ Divisor = Quotient. We can also write this as: Dividend = Divisor $\times$ Quotient. Let us look at how the divisor affects the quotient.

Step 5 — Creating the table

We will use different dividends and divisors. We will see the quotient and its relation to the dividend.

| Dividend | Divisor | Quotient | Relationship (Quotient vs. Dividend) | | :------- | :------ | :------- | :----------------------------------- | | 10 | 0.5 | 20 | Quotient is greater than Dividend | | 10 | 1 | 10 | Quotient is equal to Dividend | | 10 | 2.5 | 4 | Quotient is smaller than Dividend | | 5 | 0.2 | 25 | Quotient is greater than Dividend | | 5 | 1 | 5 | Quotient is equal to Dividend | | 5 | 1.25 | 4 | Quotient is smaller than Dividend |

When the divisor is less than 1, the quotient is greater than the dividend. When the divisor is equal to 1, the quotient is equal to the dividend. When the divisor is greater than 1, the quotient is smaller than the dividend.

Answer

(i) No, the quotient is not always greater than the dividend when the divisor is a decimal. (ii) The relationship between dividend, divisor, and quotient is: Dividend $\div$ Divisor = Quotient. If the divisor is less than 1, the quotient is greater than the dividend. If the divisor is equal to 1, the quotient is equal to the dividend. If the divisor is greater than 1, the quotient is smaller than the dividend. (iii) The table below shows the relationship:

| Dividend | Divisor | Quotient | Relationship (Quotient vs. Dividend) | | :------- | :------ | :------- | :----------------------------------- | | 10 | 0.5 | 20 | Quotient is greater than Dividend | | 10 | 1 | 10 | Quotient is equal to Dividend | | 10 | 2.5 | 4 | Quotient is smaller than Dividend | | 5 | 0.2 | 25 | Quotient is greater than Dividend | | 5 | 1 | 5 | Quotient is equal to Dividend | | 5 | 1.25 | 4 | Quotient is smaller than Dividend |