Answer:
a) 12 : 5
b) 3 : 16 : 2
c) 1520 : 1423 : 1600
d) 16 : 16 : 3
e) 7 : 12
Explanation:
You want the given ratios to be expressed as a ratio of integers reduced to lowest terms.
- 72 : 30
- 12 : 64 : 8
- 1.52 : 1.432 : 1.6
- 5/6 : 10/12 : 5/32
- 1 2/3 : 2 6/7
Ratios
The desired ratios will be found by dividing each element in the ratio by the greatest common factor (GCF) of the elements. Euclid's algorithm is useful for finding the GCF of non-integer numbers. When more than one number is involved, the GCF can be found by considering them two at a time: GCF(a, b, c, ...) = GCF(GCF(a, b), c, ...).
It can be useful to eliminate fractions at the start by multiplying by a suitable common denominator.
72 : 30
By Euclid's algorithm, the GCF is ...
72/30 = 2 r 12
30/12 = 2 r 6
12/6 = 2 r 0 . . . . . 6 is the GCF
{72, 30}/6 = {12, 5}
The reduced ratio is 12 : 5.
12 : 64 : 8
By Euclid's algorithm, the GCF is ...
64/12 = 5 r 4
12/4 = 3 r 0 . . . . . 4 is the GCF of 12 and 64
8/4 = 2 r 0 . . . . . . 4 is the GCF of 4 and 8
{12, 64, 8}/4 = {3, 16, 2}
The reduced ratio is 3 : 16 : 2.
1.52 : 1.423 : 1.6
Multiplying by 1000 to eliminate fractions, we have ...
1520 : 1423 : 1600
We note that 1423 is a prime number, so the GCF of these values is 1.
The reduced ratio is 1520 : 1423 : 1600.
5/6 : 10/12 : 5/32
We note right away that 10/12 can be reduced to 5/6. Multiplying these fractions by 96, we have ...
{5/6, 5/6, 5/32}·96 = {80, 80, 15}
The GCF of 80 and 15 is found to be ...
80/15 = 5 r 5
15/5 = 3 r 0 . . . . . . 5 is the GCF
{80, 80, 15}/5 = {16, 16, 3}
The reduced ratio is 16 : 16 : 3.
1 2/3 : 2 6/7
Multiplying by 3·7 = 21, we have ...
{1 2/3, 2 6/7}·21 = {35, 60}
The GCF of 35 and 60 is ...
60/35 = 1 r 25
35/25 = 1 r 10
25/10 = 2 r 5
10/5 = 2 r 0 . . . . . 5 is the GCF
{35, 60}/5 = {7, 12}
The reduced ratio is 7 : 12.
Note that this can also be found by dividing one number by the other:
(1 2/3)/(2 6/7) = (5/3)/(20/7) = (5/3)·(7/20) = (5/20)·(7/3) = (1/4)(7/3) = 7/12
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Additional comment
Euclid's algorithm for finding the GCF can be summarized as ...
- divide the largest number by the smallest
- if the remainder is 0, the smallest is the GCF
- otherwise, replace the largest by the remainder and repeat.
This works with any rational numbers, not restricted to integers. It can be a useful way to find the GCF of mixed numbers or fractions.
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