A​ quantity, its two thirds

​, its one fourth
​, and its one fifth
​, added​ together, become 37.
What is the​ quantity?

1 answer

Let the quantity be \( x \).

According to the information given in the problem, we can set up the following equation:

\[ x + \frac{2}{3}x + \frac{1}{4}x + \frac{1}{5}x = 37 \]

Now, we will find a common denominator for the fractions. The least common multiple of 3, 4, and 5 is 60. We will express each term with a denominator of 60:

\[ x = \frac{60}{60}x \] \[ \frac{2}{3}x = \frac{40}{60}x \] \[ \frac{1}{4}x = \frac{15}{60}x \] \[ \frac{1}{5}x = \frac{12}{60}x \]

Now, we can rewrite the equation:

\[ \frac{60}{60}x + \frac{40}{60}x + \frac{15}{60}x + \frac{12}{60}x = 37 \]

Combining the fractions on the left side:

\[ \frac{60 + 40 + 15 + 12}{60}x = 37 \]

Calculating the numerator:

\[ 60 + 40 + 15 + 12 = 127 \]

So, we have:

\[ \frac{127}{60}x = 37 \]

Now, to isolate \( x \), multiply both sides by \( \frac{60}{127} \):

\[ x = 37 \times \frac{60}{127} \]

Calculating this value:

\[ x = \frac{2220}{127} \approx 17.4418 \]

To express the quantity as a fraction or decimal:

Calculating \( 37 \times 60 \):

\[ 37 \times 60 = 2220 \]

Now divide by 127:

\[ x \approx 17.4418 \]

Thus, the exact value of the quantity is:

\[ x = \frac{2220}{127} \]

The approximate value is \( x \approx 17.44 \). However, if you need the exact fraction, you can leave it as \( \frac{2220}{127} \).