Cross multiplication
In a proportion, the two crossing products are equal.
3 : 4 and 6 : 8
3 : 4 and 6 : 8 are in proportion. Multiply the terms that cross: 3 × 8 along one diagonal and 4 × 6 along the other.
How do the two crossing products compare?
What this lesson covers
The idea
If a : b :: c : d, then ad = bc, since c = fa and d = fb for the same factor f; two ratios are proportional if their terms are equal when cross multiplied.
3 : 4 and 6 : 8
3 : 4 and 6 : 8 are in proportion. Multiply the terms that cross: 3 × 8 along one diagonal and 4 × 6 along the other.
How do the two crossing products compare?
- They are equal
- 3 × 8 is bigger
- 4 × 6 is bigger
Build a proportion, then break it
Pick a, b and a factor. Then c = factor × a and d = factor × b. Look at the two diagonal products. Then nudge d by 1.
The crossing products are equal
If a : b :: c : d, then ad = bc, since c = fa and d = fb for the same factor f; two ratios are proportional if their terms are equal when cross multiplied.
ad is a × fb and bc is b × fa. Both are the same three numbers a, b and f multiplied, so ad = bc.
For 3 : 4 :: 6 : 8: 3 × 8 = 24 and 4 × 6 = 24.
Notes
If a : b :: c : d, then ad = bc, since c = fa and d = fb for the same factor f; two ratios are proportional if their terms are equal when cross multiplied.
Check yourself
Cross-multiply to find the pair that is in proportion.
6 : 9 and 8 : 12 are in proportion. What is the common value of the two cross products?
Answer: 72
6 × 12 = 72 and 9 × 8 = 72. Equal, as they must be in a proportion.
If a : b :: c : d, which statement must be true?
Meena checks 5 : 7 and 15 : 22. She finds 5 × 22 = 110 and 7 × 15 = 105. What does she conclude?
- 3 : 5 and 6 : 11. 3 × 11 = 33 but 5 × 6 = 30. Not equal.
- 4 : 9 and 12 : 27 — correct. Yes! 4 × 27 = 108 and 9 × 12 = 108. Equal, so they are in proportion.
- 5 : 6 and 15 : 20. 5 × 20 = 100 but 6 × 15 = 90. Not equal.
- a + d = b + c. Try 3 : 4 :: 6 : 8. Then 3 + 8 = 11 but 4 + 6 = 10.
- ab = cd. Try 3 : 4 :: 6 : 8. Then 3 × 4 = 12 but 6 × 8 = 48.
- ac = bd. Try 3 : 4 :: 6 : 8. Then 3 × 6 = 18 but 4 × 8 = 32.
- ad = bc — correct. Yes! The crossing products are equal: a × d and b × c.
- They are in proportion, because both products are close. Close is not equal. Proportion needs the products to be exactly equal.
- They are in proportion, because 110 is bigger. Proportion needs the two products to be equal, not one bigger.
- They are not in proportion, because 110 ≠ 105 — correct. Yes! The crossing products are not equal.