1. If a and b are two #'s or quantities and b ≠ 0, then
the ratio of a to b is a b
An equation that states two ratios are equal. a b c d =
In the proportion
, b and c are
the means and a and d are the extremes.
The geometric mean of two positive numbers, a and b,
is the positive number, x, that satisfies a x b =

2. 76cm
8cm
76cm
8cm 19 2
12in
1ft 48 24 2 1
or just 2

3. 3 5 4 16 ft ft 12 20
2ft ft
12ft ft ft2

4. 4:18 = 2:9 4 18 2 9 =
11:1 or 11 1
= 11
2l + 2w = P
2(5x) + 2(2x) = 21
10x + 4x = 21
14x = 21
x = 1.5
l = 5x
l = 5(1.5)
l = 7.5ft
w = 2x
w = 2(1.5)
w = 3ft

5. 9 20 9 ° ° °
x + 4x + 5x = 180
10x = 180
x = 18
18°, 72°, 90°

6. ad bc

7. 12 4 2 2x

8. 117 3 n 24 n
2808 = 3n
936

9. ab ab

10. Any time you have a number of factors contributing to a product, and you want to find the "average" factor, the answer is the geometric mean. The example of interest rates is probably the application most used in everyday life.
For example, suppose you have an investment which earns 10% the first year, 60% the second year, and 20% the third year. What is its average rate of return? It is not the arithmetic mean, because what these numbers mean is that on the first year your investment was multiplied (not added to) by 1.10, on the second year it was multiplied by 1.60, and the third year it was multiplied by The relevant quantity is the geometric mean of these three numbers.
The question about finding the average rate of return can be rephrased as: "by what constant factor would your investment need to be multiplied by each year in order to achieve the same effect as multiplying by 1.10 one year, 1.60 the next, and 1.20 the third?" The answer is the geometric mean . If you calculate this geometric mean you get approximately 1.283, so the average rate of return is about 28% (not 30% which is what the arithmetic mean of 10%, 60%, and 20% would give you). x = ∛(1.10×1.60×1.20)
It is known that the geometric mean is always less than or equal to the arithmetic mean (equality holding only when A=B). The proof of this is quite short and follows from the fact that is always a non-negative number. This inequality can be surprisingly powerful though and comes up from time to time in the proofs of theorems in calculus.

11. √ab
√16×48
√768
√256×3 or √16×16×3
16√3
16√

12. 9(x - 3) = 6x 40 = 2y 9x - 27 = 6x 20 = y -27 = -3x 9 = x x = √ab10 x
192 3 =
640 = x 1 64
x = √ab
x = √14×16
x = 4√14 or
x ≈