Question

Column A: \(\frac{13}{15} + \frac{7}{8} + \frac{3}{4}\)
Column B: 3

• A. Quantity A is greater
• B. Quantity B is greater
• C. The two quantities are equal
• D. The relationship cannot be determined from the information given

Hint:

When comparing a sum of fractions to an integer, first try estimation. If each fraction is less than 1, their sum will be less than the number of fractions. This can often save you from complex calculations.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question asks to compare the sum of three fractions to an integer. We can solve this either by finding the exact sum or by using estimation.
Step 2: Key Formula or Approach:
Method 1: Estimation
We can compare each fraction to 1.
\(\frac{13}{15}\) is less than 1 (since \(13 \textless 15\)). It is \(1 - \frac{2}{15}\).
\(\frac{7}{8}\) is less than 1 (since \(7 \textless 8\)). It is \(1 - \frac{1}{8}\).
\(\frac{3}{4}\) is less than 1 (since \(3 \textless 4\)). It is \(1 - \frac{1}{4}\).
The sum is the sum of three numbers, each of which is less than 1. Therefore, their sum must be less than \(1 + 1 + 1 = 3\).
This quick estimation shows that the quantity in Column A is less than the quantity in Column B.
Method 2: Exact Calculation
To find the exact sum, we need to find a common denominator for 15, 8, and 4.
The multiples of 15 are 15, 30, 45, 60, 75, 90, 105, 120, ...
The multiples of 8 are 8, 16, ..., 120, ...
The multiples of 4 are 4, 8, ..., 120, ...
The least common multiple (LCM) of 15, 8, and 4 is 120.
Now, convert each fraction to an equivalent fraction with a denominator of 120.
\[ \frac{13}{15} = \frac{13 \times 8}{15 \times 8} = \frac{104}{120} \] \[ \frac{7}{8} = \frac{7 \times 15}{8 \times 15} = \frac{105}{120} \] \[ \frac{3}{4} = \frac{3 \times 30}{4 \times 30} = \frac{90}{120} \] Now, add the fractions:
\[ \frac{104}{120} + \frac{105}{120} + \frac{90}{120} = \frac{104 + 105 + 90}{120} = \frac{299}{120} \] Step 3: Comparing the Quantities:
Column A: \(\frac{299}{120}\)
Column B: 3
To compare, we can write 3 as a fraction with a denominator of 120.
\[ 3 = \frac{3 \times 120}{120} = \frac{360}{120} \] Comparing the numerators, we see that \(299 \textless 360\).
Therefore, \(\frac{299}{120} \textless 3\).
Step 4: Final Answer:
The quantity in Column A is less than the quantity in Column B. Both methods confirm this result.