# Simplify ratio 152:121.6:212.8

How am I supposed to know that I can simplify the ratio by dividing with 30.4? Is that such an obvious common factor? How would you go about this if you had 60 seconds time to solve it?

You're supposed to arrive at 5:4:7.

• Multiply each number by 10 to get the decimals out of the way. Then start dividing by 2's, then possibly 3's etcetera until it can't go any further – imranfat May 5 '14 at 21:15
• @imranfat Until it comes the moment you need to notice the factor $19$. Then, you will need some training to get it in the 60 seconds mark. – user147444 May 5 '14 at 21:18
• that's what I thought.. – TMOTTM May 5 '14 at 21:21
• The 19 comes quickly if you factor the numbers. Or you could use the Euclidean algorithm to compute the GCD after the obvious simplifications. – Hurkyl May 5 '14 at 21:21
• That happens when you arrive at 85:76:133 , still doable but your point is well taken. It's just that 85 and 76 are recognizable multiples of 19 and so that would be a suitable try... – imranfat May 5 '14 at 21:22

Take $152:121.6:212.8$

Multiply by five to remove decimals: $760:608:1064$

Take out a multiple of $2^3=8$ to get $95:76:133$

$95=5\times 19$ is an obvious factorisation. $5$ is clearly not a common factor. $19$ turns out to be, and gives $$5:4:7$$

• Alternatively $76 = 2\times 38=2\times 2 \times 19$ or you know (as I do, by accident) $133 = 7 \times 19$. Once you have identified $19$ as a possible factor at any stage, the problem becomes easy. – Mark Bennet May 5 '14 at 21:53

I would do $$\frac{152}{121.6}=1.25=\frac54, \ \ \frac{121.6}{212.8}=0.\overline{5714285}=\frac{571428}{999999}=\frac47.$$

Or if you cannot recognize the second fraction right away, $$\frac{121.6}{212.8}=\frac{1216}{2128}=\frac{64\times19}{16\times133}=\frac{64\times19}{16\times7\times19}=\frac47.$$