Which of the following pairs of rational numbers are equivalent? 18 46 (0) 14:42 () -27 -6
The Correct Answer and Explanation is:
Correct Answer: -27 and -6
To determine which pairs of rational numbers are equivalent, simplify each pair of numbers by expressing them as fractions and reducing them to their lowest terms.
1. First Pair: 18 and 46
Express these numbers as a fraction:1846\frac{18}{46}4618
Find the greatest common divisor (GCD) of 18 and 46.
The GCD is 2. Divide both numerator and denominator by 2:18÷246÷2=923\frac{18 \div 2}{46 \div 2} = \frac{9}{23}46÷218÷2=239
Since 923\frac{9}{23}239 is in its simplest form, it cannot be simplified further. It does not equal 1 or any commonly known simplified fraction equivalent to a whole number or another rational number in this list.
2. Second Pair: 14 and 42
Express these numbers as a fraction:1442\frac{14}{42}4214
The GCD of 14 and 42 is 14. Simplify the fraction:14÷1442÷14=13\frac{14 \div 14}{42 \div 14} = \frac{1}{3}42÷1414÷14=31
This results in a simplified fraction 13\frac{1}{3}31, which again does not equal 1 or any fraction equivalent to -27 or -6. So, this pair is not equivalent to any others.
3. Third Pair: -27 and -6
Write the two numbers as a fraction:−27−6\frac{-27}{-6}−6−27
Dividing a negative number by another negative number results in a positive value:−27−6=276\frac{-27}{-6} = \frac{27}{6}−6−27=627
Now simplify 276\frac{27}{6}627. The GCD of 27 and 6 is 3.27÷36÷3=92\frac{27 \div 3}{6 \div 3} = \frac{9}{2}6÷327÷3=29
Since the simplified fraction is 92\frac{9}{2}29, the pair is equivalent in value — both values can be represented as 92\frac{9}{2}29.
Conclusion:
The only equivalent pair of rational numbers is -27 and -6, because their ratio simplifies to the same positive rational number.
