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3 years ago

Compare fractions 3/4 and 11/12

Mathematics

1 answer:

RoseWind [281]3 years ago

5 0

For now, assume 3/4 and 11/12 are equal. It is likely they are not equal, but we'll get to that later.

The equation we set up is 3/4 = 11/12

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3/4 = 11/12

3*12 = 4*11 ... cross multiply

36 = 44

The last equation is false, so the entire string of equations are false.

This tells us that 3/4 is not equal to 11/12.

Since the left hand side of the last equation is smaller, this means we must replace the equal sign with a less than sign. Do so with the other equal signs as well. We have this updated set of steps

3/4 < 11/12

3*12 < 4*11 ... cross multiply

36 < 44

We see that 3/4 is smaller than 11/12

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Another way to see that 3/4 is smaller is to use a calculator to find the decimal version of each fraction

3/4 = 0.75

11/12 = 0.9167 approximately

Since 0.75 is smaller than 0.9167, this means 3/4 is smaller than 11/12.

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If you want to stick with fractions and you don't want to cross multiply, then we can rewrite 3/4 as a fraction that has 12 in the denominator.

Multiply top and bottom of 3/4 by the value 3

This will make 3/4 turn into 9/12

3*3 = 9

4*3 = 12

So 3/4 = 9/12 is a true equation

Instead of comparing "3/4 and 11/12", we are equivalently comparing "9/12 and 11/12".

The numerator 9 is smaller than the numerator 11, so 9/12 is smaller than 11/12

This points to 3/4 being smaller than 11/12.

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Now we will switch sides:

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Let us take natural log on both sides of equation.

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$-0.05t\cdot (1)=\text{ln}(\frac{1}{3})$

$-0.05t=\text{ln}(\frac{1}{3})$

$t=\frac{\text{ln}(\frac{1}{3})}{-0.05}$

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$t=-(0-\text{ln}(3))\cdot 20$

$t=20\text{ln}(3)$

Therefore, it will take $20\text{ln}(3)$ years for area of the glacier to decrease to 15 square kilo-meters.

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