(A) 17/18
(B) 28/36
(C) 43/45
(D) 59/60
Choose the correct answer from the options given below:
Fractions Denominators Equal Explained
The question asks us to identify which of the given fractions, after converting them to have equal denominators, will result in a numerator that is an even number.
Denominators Equal Common Multiple
To compare or order fractions, a common approach is to convert them to equivalent fractions with a common denominator. The least common multiple (LCM) of the original denominators provides the smallest common denominator.
The denominators of the given fractions are 18, 36, 45, and 60.
Let's find the LCM of these denominators:
- Prime factorization of 18: \( 2 \times 3^2 \)
- Prime factorization of 36: \( 2^2 \times 3^2 \)
- Prime factorization of 45: \( 3^2 \times 5 \)
- Prime factorization of 60: \( 2^2 \times 3 \times 5 \)
The LCM is calculated by taking the highest power of each prime factor present in any of the denominators:
LCM = \( 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180 \)
Thus, the common denominator for all fractions will be 180.
Fractions Conversion Numerator Parity
Now, we will convert each fraction into an equivalent fraction with the denominator 180 and examine the parity (even or odd) of the resulting numerator.
| Fraction (Option) | Calculation to achieve Denominator 180 | Equivalent Fraction | Resulting Numerator | Parity of Numerator |
|---|---|---|---|---|
| (A) 17/18 | \( \frac{17}{18} \times \frac{180 \div 18}{180 \div 18} = \frac{17}{18} \times \frac{10}{10} \) | 170/180 | 170 | Even |
| (B) 28/36 | \( \frac{28}{36} \times \frac{180 \div 36}{180 \div 36} = \frac{28}{36} \times \frac{5}{5} \) | 140/180 | 140 | Even |
| (C) 43/45 | \( \frac{43}{45} \times \frac{180 \div 45}{180 \div 45} = \frac{43}{45} \times \frac{4}{4} \) | 172/180 | 172 | Even |
| (D) 59/60 | \( \frac{59}{60} \times \frac{180 \div 60}{180 \div 60} = \frac{59}{60} \times \frac{3}{3} \) | 177/180 | 177 | Odd |
Conclusion Numerator Parity Result
The analysis shows that when the fractions are converted to have the common denominator 180:
- Fraction (A) 17/18 becomes 170/180, with an even numerator.
- Fraction (B) 28/36 becomes 140/180, with an even numerator.
- Fraction (C) 43/45 becomes 172/180, with an even numerator.
- Fraction (D) 59/60 becomes 177/180, with an odd numerator.
Therefore, the fractions where the numerators will be an even number after making the denominators equal are (A), (B), and (C).
Important Questions from Number System (Notes)
- The sum of the digits of a two-digit number is 10. If 18 is subtracted from it, the digits in the resulting number will be equal. The number is :
- A 6-digit number has digits as consecutive natural numbers. The number is always divisible by :
- Find the number of trailing zeros in $15620!$
- The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they all change simultaneously at 8:20:00 hours, then at what time will they again change simultaneously?
- The number 1071819 is divisible by:
