SOLUTION: a year ago, the daily sales of a newspaper averaged 16 320.Now the average daily sales are 28 560. Express the ratio of present sales to last year's sales in its simplest terms

Question 1143844: a year ago, the daily sales of a newspaper averaged 16 320.Now the average daily sales are 28 560. Express the ratio of present sales to last year's sales in its simplest terms
Found 2 solutions by ikleyn, KMST:
Answer by ikleyn(52848) (Show Source): You can put this solution on YOUR website!
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Just solved under this link

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https://www.algebra.com/algebra/homework/Percentage-and-ratio-word-problems/Percentage-and-ratio-word-problems.faq.question.1143845.html

Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Dividing both numbers by a common divisor you get an equivalent ratio in smaller term. to get the simplest ratio you need to use the greatest common divisor.

Both numbers (16 320 and 28 480) and their difference (12 240) are divisible by 4080.
16 320 divided by 4 080 is 4.
28 560 divided by 4 080 is 7.
The ratio 28 560 : 16 320 can be expressed as 7:4 in simplest terms.
The ratio 7:4 is in simplest terms because 7 and 4 do not have any common factor, other than 1.

If you can see right away that both numbers divide by 4 080 evenly, you do not need to do any more work. If you do not see that divisor, you can divide both number by common divisors repeatedly, or you can find the their prime number factorizations and from there calculate the greatest common divisor.
For example, to find the prime factorization of 12 240, you could to start divide it by 2 repeatedly, finding 6120, 3040, 1530, and 765. Next you try dividing by 3 as many times as you get a whole number result, getting 255, and 85. Then, you would try dividing 85 by the next prime number, which is 5, and would find the result is 17, which is a prime number. so, the prime factorization of 12 240 is
12 240 = 2 X 2 X 2 X 2 X 3 X 3 X 5 X 17.

TRICK:
If two numbers have a common divisor, it is also a divisor of their difference, but not all divisors of the difference are common divisors.

I took a quick look at the difference (12 240), and thought that number would be easy to work with. Dividing by 10, you get 1 224, which is divisible by 8, just like 1632 and 2856. 1224 is divisible by 9=3x3, but 1632 and 2856 are not divisible by 9, only by 3. SO far I found that I can divide both numbers by 10, 8, and 3, which means they divide by 3x8x10=240.
Dividing 16 320 by 10, I got 1632. Dividing that by 8, I got 204, and dividing 204 by 3, I got 68.
68 = 4 X 17.
As both numbers are divisible by 10, 8, and 3, and 8, if they were both divisible by 17 they would be divisible by 3X17=51, by 8X51=408, and by 408X10=4080.
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