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The ratio of the areas of two squares is 192/80. After rationalizing the denominator, the ratio of their side lengths can be expressed in the simplified form $\frac{a\sqrt{b}}{c}$ where a, b, and c are integers. What is the value of the sum a+b+c?

Jun 11, 2019

#1
+8437
+3

The ratio of their side lengths is...

$$\sqrt{\frac{192}{80}}\ =\ \sqrt{\frac{12}{5}}\ =\ \frac{\sqrt{12}}{\sqrt5}\ =\ \frac{\sqrt{12}\,\cdot\,\sqrt5}{\sqrt5\,\cdot\,\sqrt5}\ =\ \frac{\sqrt{12\,\cdot\,5}}{\sqrt{5\,\cdot\,5}}\ =\ \frac{\sqrt{2\,\cdot\,2\,\cdot\,3\,\cdot\,5}}{\sqrt{5\,\cdot\,5}}\ =\ \frac{\sqrt{2\,\cdot\,2}\,\cdot\,\sqrt{3\,\cdot\,5}}{\sqrt{5\,\cdot\,5}}\ =\ \frac{2\sqrt{15}}{5}$$

Now it is in the simplified form  $$\frac{a\sqrt{b}}{c}$$  where  a,  b,  and  c  are integers.

a + b + c  =  2 + 15 + 5  =  22

Jun 11, 2019

#1
+8437
+3

The ratio of their side lengths is...

$$\sqrt{\frac{192}{80}}\ =\ \sqrt{\frac{12}{5}}\ =\ \frac{\sqrt{12}}{\sqrt5}\ =\ \frac{\sqrt{12}\,\cdot\,\sqrt5}{\sqrt5\,\cdot\,\sqrt5}\ =\ \frac{\sqrt{12\,\cdot\,5}}{\sqrt{5\,\cdot\,5}}\ =\ \frac{\sqrt{2\,\cdot\,2\,\cdot\,3\,\cdot\,5}}{\sqrt{5\,\cdot\,5}}\ =\ \frac{\sqrt{2\,\cdot\,2}\,\cdot\,\sqrt{3\,\cdot\,5}}{\sqrt{5\,\cdot\,5}}\ =\ \frac{2\sqrt{15}}{5}$$

Now it is in the simplified form  $$\frac{a\sqrt{b}}{c}$$  where  a,  b,  and  c  are integers.

a + b + c  =  2 + 15 + 5  =  22

hectictar Jun 11, 2019
#2
+248
+1

Thanks!

sinclairdragon428  Jun 11, 2019