Arrange the fractions $\frac{17}{18}$, $\frac{31}{36}$, $\frac{43}{45}$, $\frac{59}{60}$ in the ascending order.
Aptitude
HCF and LCM
Difficulty: Medium
Choose an option
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A$\frac{31}{36} < \frac{17}{18} < \frac{43}{45} < \frac{59}{60}$
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B$\frac{17}{18} < \frac{31}{36} < \frac{43}{45} < \frac{59}{60}$
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C$\frac{31}{36} < \frac{43}{45} < \frac{17}{18} < \frac{59}{60}$
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D$\frac{31}{36} < \frac{17}{18} < \frac{59}{60} < \frac{43}{45}$
Answer
Correct Answer: $\frac{31}{36} < \frac{17}{18} < \frac{43}{45} < \frac{59}{60}$
Explanation
### Concept & Strategy
To reliably compare and order multiple fractions, convert them into equivalent fractions with a common denominator. The common denominator will be the L.C.M. of the original denominators. Once the denominators are identical, simply sort the fractions based on the values of their numerators.
### Step-by-Step Solution
* **Given:** The fractions are $\frac{17}{18}$, $\frac{31}{36}$, $\frac{43}{45}$, and $\frac{59}{60}$.
* Find the L.C.M. of the denominators (18, 36, 45, and 60).
* L.C.M. of 18, 36, 45, 60 = 180.
* Convert each fraction to an equivalent fraction with a denominator of 180:
* $\frac{17}{18} = \frac{17 \times 10}{18 \times 10} = \frac{170}{180}$
* $\frac{31}{36} = \frac{31 \times 5}{36 \times 5} = \frac{155}{180}$
* $\frac{43}{45} = \frac{43 \times 4}{45 \times 4} = \frac{172}{180}$
* $\frac{59}{60} = \frac{59 \times 3}{60 \times 3} = \frac{177}{180}$
* Compare the new numerators:
* $155 < 170 < 172 < 177$
* Therefore, their corresponding equivalent fractions follow the same ascending order:
* $\frac{155}{180} < \frac{170}{180} < \frac{172}{180} < \frac{177}{180}$
* Substitute back the original fractions:
* $\frac{31}{36} < \frac{17}{18} < \frac{43}{45} < \frac{59}{60}$
### Exam Strategy & Shortcut
Use the **Numerator-Denominator Difference Trick**. Look at the gap between the numerator and denominator for each fraction:
* $\frac{17}{18}$ (Gap = 1)
* $\frac{31}{36}$ (Gap = 5)
* $\frac{43}{45}$ (Gap = 2)
* $\frac{59}{60}$ (Gap = 1)
For proper fractions where values are close to 1, the fraction with the largest gap is generally the smallest. The gap of 5 implies $\frac{31}{36}$ is the smallest. The gap of 2 implies $\frac{43}{45}$ is next. The gap of 1 is shared by two fractions, so compare them directly: $\frac{17}{18}$ vs $\frac{59}{60}$. Since 59/60 is closer to 60/60, it is larger.
Result: $\frac{31}{36} < \frac{17}{18} < \frac{43}{45} < \frac{59}{60}$.
### Common Pitfall
Attempting to convert all fractions into exact decimals by long division under time pressure is a major mistake. It takes far too much time and increases the risk of calculation errors. Stick to L.C.M. equivalence or gap observation methods.
### Final Answer
**Therefore, the correct answer is $\frac{31}{36} < \frac{17}{18} < \frac{43}{45} < \frac{59}{60}$.**