Let the unknown quantity be \( x \).
According to the problem, the equation can be set up as follows:
\[ x + \frac{3}{4}x + \frac{1}{3}x + \frac{1}{5}x = 89 \]
To solve this, we first need a common denominator for the fractions. The denominators are 1, 4, 3, and 5. The least common multiple of these numbers is 60.
We can rewrite the equation using the common denominator:
\[ x + \frac{3}{4}x + \frac{1}{3}x + \frac{1}{5}x = 89 \]
Express each term with a denominator of 60:
- \( x = \frac{60}{60}x \)
- \( \frac{3}{4}x = \frac{45}{60}x \) (because \( 3 \times 15 = 45 \))
- \( \frac{1}{3}x = \frac{20}{60}x \) (because \( 1 \times 20 = 20 \))
- \( \frac{1}{5}x = \frac{12}{60}x \) (because \( 1 \times 12 = 12 \))
Now substitute these back into the equation:
\[ \frac{60}{60}x + \frac{45}{60}x + \frac{20}{60}x + \frac{12}{60}x = 89 \]
Combine the fractions:
\[ \frac{60 + 45 + 20 + 12}{60}x = 89 \]
Calculate the sum in the numerator:
\[ 60 + 45 + 20 + 12 = 137 \]
So, the equation now looks like this:
\[ \frac{137}{60}x = 89 \]
To solve for \( x \), multiply both sides by the reciprocal of \(\frac{137}{60}\):
\[ x = 89 \times \frac{60}{137} \]
Calculating this gives:
\[ x = \frac{5340}{137} \approx 39.0 \]
To check, we can substitute \( x = 39 \) back into the original terms:
- \( x = 39 \)
- \( \frac{3}{4} \times 39 = 29.25 \)
- \( \frac{1}{3} \times 39 = 13 \)
- \( \frac{1}{5} \times 39 = 7.8 \)
Now sum these values:
\[ 39 + 29.25 + 13 + 7.8 = 89.05 \quad (\text{approximately } 89) \]
Thus, the original quantity \( x \) is approximately 39.