22
The basic property of a proportion
Textbook, pp. 93–97
GoalKnow the basic property of a proportion; use it to check proportions and find an unknown term.
New words
basic property · asosiy xossaproduct of the extremes · chekka hadlar ko‘paytmasiproduct of the means · o‘rta hadlar ko‘paytmasiunknown term · noma’lum had
Explanation
The basic property of a proportion: the product of the extremes equals the product of the means; that is, if a : b = c : d then a · d = b · c. The converse is also true: if the products are equal, the proportion holds. We use it to check: 3 : 5 = 6 : 10 because 3 · 10 = 30 and 5 · 6 = 30. To find an unknown extreme, divide the product of the means by the known extreme; to find an unknown mean, divide the product of the extremes by the known mean.
Worked examples
4 : 7 = 12 : 20? Extremes: 4 · 20 = 80, means: 7 · 12 = 84. Since 80 ≠ 84, it is not a proportion.
In x : 4 = 9 : 6, x is an extreme: x = 4 · 9 : 6 = 6.
Class activity
“Cross-product check”: each team gets cards with proportions, some of them false. Teams calculate the products of extremes and means and pick out the false ones.
Practice
1
Check the proportion 6 : 9 = 8 : 12.
True: 6 · 12 = 72 and 9 · 8 = 72.
2
Is 4 : 7 = 12 : 20 a proportion?
No: 4 · 20 = 80 but 7 · 12 = 84.
3
In x : 4 = 9 : 6, find x.
6
4
Why do we divide the product of the means by the known extreme to find an unknown extreme?
From a · d = b · c, divide both sides by d to get a = (b · c) : d.