Comparing the quotient with the dividend

The idea

When the divisor is between 0 and 1, the quotient is greater than the dividend; when the divisor is greater than 1, the quotient is less than the dividend.

Is the quotient always smaller?

Divide whole numbers: 6 ÷ 3 = 2. The quotient 2 is smaller than the dividend 6. Now divide 6 by a part of one: 6 ÷ 1/4.

How many quarters are there in 6? So 6 ÷ 1/4 is…

• 24
• 3/2
• 2/3

Move the numbers, fill the table

Drag x (the dividend) and y (the divisor), or tap ◀ ▶. The green marker is the quotient x ÷ y. Find out what happens in each box of the table.

Divide by a number below 1: it grows

Cut something into pieces smaller than 1 and you get more pieces than you started with: the quotient is bigger. Cut it into pieces bigger than 1 and you get fewer: the quotient is smaller. Only the divisor decides.

When the divisor is between 0 and 1, the quotient is greater than the dividend. When the divisor is greater than 1, the quotient is less than the dividend.

• Division | Quotient
• 6 ÷ 3 | 2: smaller than 6 (the divisor is above 1)
• 6 ÷ 1/4 | 24: bigger than 6 (the divisor is below 1)
• 1/8 ÷ 1/4 | 1/2: bigger than 1/8 (the divisor is below 1)

Notes

When the divisor is between 0 and 1, the quotient is greater than the dividend. When the divisor is greater than 1, the quotient is less than the dividend.

Check yourself

Without working it out: 9 ÷ 3/4 is…

Which division has a quotient less than 5?

Find 5/2 ÷ 5/4. The divisor is above 1, so the answer must be less than 5/2. Type a whole number.

Answer: 2

5/2 × 4/5 = 20/10 = 2, which is less than 5/2.

Ravi says: “Dividing always makes a number smaller.” Which example shows he is wrong?

• less than 9. Dividing does not always make a number smaller. A divisor below 1 makes it bigger.
• more than 9 — correct. Yes! The divisor 3/4 is between 0 and 1, so the quotient is greater than the dividend. It is 12.
• equal to 9. Only dividing by 1 leaves a number unchanged. 3/4 is less than 1.
• 5 ÷ 4/3 — correct. Yes! The divisor 4/3 is greater than 1, so the quotient is less than 5. It is 15/4.
• 5 ÷ 2/3. The divisor 2/3 is below 1, so the quotient is greater than 5. It is 15/2.
• 5 ÷ 1/10. The divisor 1/10 is below 1, so the quotient is greater than 5. It is 50.
• 6 ÷ 3 = 2. Here the quotient is smaller than 6, which fits what Ravi says.
• 3/4 ÷ 2 = 3/8. Here the quotient is smaller than 3/4, which fits what Ravi says.
• 1/8 ÷ 1/4 = 1/2 — correct. Yes! 1/2 is greater than 1/8: the divisor 1/4 is below 1.