Swap it, share it

The idea

Addition and multiplication of rational numbers are both commutative, and they follow the distributive law: for rational numbers p, q, r, p(q + r) = pq + pr.

Swap and see

You know that 3 + 5 = 5 + 3. Does the same hold for fractions? And what happens if you multiply a number by a sum?

If you swap the two numbers in 3/4 + 1/2, does the sum change?

• Yes, it changes
• No, it stays the same
• Only for whole numbers

Three tests

Open each tab, press the button, and compare the totals before and after.

Order and sharing

Addition and multiplication of rational numbers are both commutative: a/b + c/d = c/d + a/b and a/b × c/d = c/d × a/b. They also follow the distributive law: p(q + r) = pq + pr.

The order does not matter when you add or when you multiply. Multiplying a sum gives the same result as multiplying each part and then adding.

Notes

Addition and multiplication of rational numbers are commutative, and they follow the distributive law: p(q + r) = pq + pr.

Check yourself

Which number is the same as 2/5 + 3/7?

5/6 + 1/4 = 13/12. By the commutative law, 1/4 + 5/6 = ?/12. What is ?

Answer: 13

1/4 + 5/6 = 5/6 + 1/4 = 13/12, so the top is 13.

Use p(q + r) = pq + pr to work out 12 × (1/2 + 1/3). 12 × 1/2 + 12 × 1/3 = 6 + 4 = ?

Answer: 10

12 × 1/2 = 6 and 12 × 1/3 = 4, so the total is 6 + 4 = 10. Check: 1/2 + 1/3 = 5/6 and 12 × 5/6 = 10.

Which one is the distributive law?

• 2/5 × 3/7. Adding and multiplying are different operations.
• 3/7 − 2/5. Subtracting is a different operation.
• 3/7 + 2/5 — correct. Yes. Addition is commutative, so the order does not matter.
• p(q + r) = pq + pr — correct. Yes. Multiplying a sum is the same as multiplying each part and adding.
• p + q = q + p. That is the commutative law for addition.
• p × q = q × p. That is the commutative law for multiplication.