Question

What is the smallest rational number among the following to be added to the following sum to make it an integer: \( 2\frac{2}{3}+4\frac{7}{12}+6\frac{5}{6}+8\frac{11}{36} \)?

• A. \( \frac{1}{18} \)
• B. \( \frac{5}{18} \)
• C. \( \frac{7}{18} \)
• D. \( \frac{11}{18} \)

Hint:

When several mixed fractions are added, first add all integer parts separately and then add the fractional parts using the LCM of denominators. The complement of the final fractional part gives the least number needed to obtain an integer.

Answer 1

The Correct Option is D

Concept: To make a number an integer, we need to examine its fractional part. If the fractional part is \(f\), then the least positive number required to make it an integer is \(1-f\).

Step 1: Separate the integer and fractional parts.
Given, \[ 2\frac{2}{3}+4\frac{7}{12}+6\frac{5}{6}+8\frac{11}{36} \] Adding the integer parts, \[ 2+4+6+8=20 \] Now add the fractional parts: \[ \frac{2}{3}+\frac{7}{12}+\frac{5}{6}+\frac{11}{36} \] Taking LCM of \(3,12,6,36\), we get \[ 36 \] Hence, \[ \frac{2}{3}=\frac{24}{36}, \qquad \frac{7}{12}=\frac{21}{36}, \qquad \frac{5}{6}=\frac{30}{36}, \qquad \frac{11}{36}=\frac{11}{36} \] Therefore, \[ \frac{24+21+30+11}{36} = \frac{86}{36} = \frac{43}{18} = 2+\frac{7}{18} \]

Step 2: Find the fractional part of the total sum.
Thus the complete sum is \[ 20+2+\frac{7}{18} = 22+\frac{7}{18} \] The fractional part is \[ \frac{7}{18} \]

Step 3: Determine the least number required.
To reach the next integer, \[ 1-\frac{7}{18} = \frac{18-7}{18} = \frac{11}{18} \] Hence the smallest rational number that must be added is \[ \boxed{\frac{11}{18}} \]