Cross and check

The idea

Two rational numbers a/b and c/d are equal if ad = bc.

A quick check

2/3 and 6/9 look different. Are they the same number? Drawing bars takes time. Is there a quick check?

Are 2/3 and 6/9 equal?

• Yes
• No
• I cannot tell without drawing bars

Multiply across

Multiply across: the top of one fraction with the bottom of the other. Set c and d with the sliders and watch the two products.

ad = bc

Two rational numbers a/b and c/d are equal if ad = bc.

Multiply the top of the first by the bottom of the second (ad). Multiply the bottom of the first by the top of the second (bc). If the two products match, the numbers are equal.

Notes

Two rational numbers a/b and c/d are equal if ad = bc.

Check yourself

Check whether 3/4 and 9/12 are equal. ad = 3 × 12 = 36. What is bc = 4 × 9?

Answer: 36

4 × 9 = 36. Both products are 36, so 3/4 = 9/12.

Which fraction is equal to 5/6?

For 4/7 and 5/9 we get ad = 4 × 9 = 36 and bc = 7 × 5 = 35. Are the two numbers equal?

Find x so that 3/5 = x/20. Then 3 × 20 = 5 × x. What is x?

Answer: 12

3 × 20 = 60 = 5 × x, so x = 60 ÷ 5 = 12. Check: 3/5 = 12/20.

• 10/12 — correct. Yes. 5 × 12 = 60 and 6 × 10 = 60.
• 10/11. 5 × 11 = 55 but 6 × 10 = 60. The products differ.
• 6/5. 5 × 5 = 25 but 6 × 6 = 36. The products differ.
• Yes, because 36 and 35 are close. Close is not equal. The products must be exactly the same.
• No, because 36 ≠ 35 — correct. Yes. The two products must match for the numbers to be equal.
• We cannot tell. We can: ad = bc is the test, and 36 ≠ 35.