Answer :
To determine which pairs of rational numbers are equal, we will use the cross-multiplication rule [tex]\(ad = bc\)[/tex]. This rule states that two fractions [tex]\( \frac{a}{b} \)[/tex] and [tex]\( \frac{c}{d} \)[/tex] are equal if and only if the cross-multiplication of their numerators and denominators (i.e., [tex]\(a \cdot d\)[/tex] and [tex]\(b \cdot c\)[/tex]) are equal.
Let's go through each pair step-by-step.
### (a) [tex]\(\frac{-12}{16}\)[/tex] and [tex]\(\frac{5}{-8}\)[/tex]
Given fractions: [tex]\(\frac{-12}{16}\)[/tex] and [tex]\(\frac{5}{-8}\)[/tex]
Here,
- [tex]\(a = -12\)[/tex]
- [tex]\(b = 16\)[/tex]
- [tex]\(c = 5\)[/tex]
- [tex]\(d = -8\)[/tex]
Now, calculate [tex]\(a \cdot d\)[/tex] and [tex]\(b \cdot c\)[/tex]:
[tex]\[ (-12) \cdot (-8) = 96 \][/tex]
[tex]\[ 16 \cdot 5 = 80 \][/tex]
Since [tex]\(96 \neq 80\)[/tex], these fractions are not equal.
### (b) [tex]\(\frac{-15}{20}\)[/tex] and [tex]\(\frac{21}{-27}\)[/tex]
Given fractions: [tex]\(\frac{-15}{20}\)[/tex] and [tex]\(\frac{21}{-27}\)[/tex]
Here,
- [tex]\(a = -15\)[/tex]
- [tex]\(b = 20\)[/tex]
- [tex]\(c = 21\)[/tex]
- [tex]\(d = -27\)[/tex]
Now, calculate [tex]\(a \cdot d\)[/tex] and [tex]\(b \cdot c\)[/tex]:
[tex]\[ (-15) \cdot (-27) = 405 \][/tex]
[tex]\[ 20 \cdot 21 = 420 \][/tex]
Since [tex]\(405 \neq 420\)[/tex], these fractions are not equal.
### (c) [tex]\(\frac{-8}{24}\)[/tex] and [tex]\(\frac{7}{-21}\)[/tex]
Given fractions: [tex]\(\frac{-8}{24}\)[/tex] and [tex]\(\frac{7}{-21}\)[/tex]
Here,
- [tex]\(a = -8\)[/tex]
- [tex]\(b = 24\)[/tex]
- [tex]\(c = 7\)[/tex]
- [tex]\(d = -21\)[/tex]
Now, calculate [tex]\(a \cdot d\)[/tex] and [tex]\(b \cdot c\)[/tex]:
[tex]\[ (-8) \cdot (-21) = 168 \][/tex]
[tex]\[ 24 \cdot 7 = 168 \][/tex]
Since [tex]\(168 = 168\)[/tex], these fractions are equal.
### (d) [tex]\(\frac{-6}{-10}\)[/tex] and [tex]\(\frac{11}{15}\)[/tex]
Given fractions: [tex]\(\frac{-6}{-10}\)[/tex] and [tex]\(\frac{11}{15}\)[/tex]
Here,
- [tex]\(a = -6\)[/tex]
- [tex]\(b = -10\)[/tex]
- [tex]\(c = 11\)[/tex]
- [tex]\(d = 15\)[/tex]
Now, calculate [tex]\(a \cdot d\)[/tex] and [tex]\(b \cdot c\)[/tex]:
[tex]\[ (-6) \cdot 15 = -90 \][/tex]
[tex]\[ (-10) \cdot 11 = -110 \][/tex]
Since [tex]\(-90 \neq -110\)[/tex], these fractions are not equal.
### Conclusion
Only one pair, pair (c), is equal:
- [tex]\(\frac{-8}{24}\)[/tex] and [tex]\(\frac{7}{-21}\)[/tex]
So, the answer is:
(c) [tex]\(\frac{-8}{24}\)[/tex] and [tex]\(\frac{7}{-21}\)[/tex].
Let's go through each pair step-by-step.
### (a) [tex]\(\frac{-12}{16}\)[/tex] and [tex]\(\frac{5}{-8}\)[/tex]
Given fractions: [tex]\(\frac{-12}{16}\)[/tex] and [tex]\(\frac{5}{-8}\)[/tex]
Here,
- [tex]\(a = -12\)[/tex]
- [tex]\(b = 16\)[/tex]
- [tex]\(c = 5\)[/tex]
- [tex]\(d = -8\)[/tex]
Now, calculate [tex]\(a \cdot d\)[/tex] and [tex]\(b \cdot c\)[/tex]:
[tex]\[ (-12) \cdot (-8) = 96 \][/tex]
[tex]\[ 16 \cdot 5 = 80 \][/tex]
Since [tex]\(96 \neq 80\)[/tex], these fractions are not equal.
### (b) [tex]\(\frac{-15}{20}\)[/tex] and [tex]\(\frac{21}{-27}\)[/tex]
Given fractions: [tex]\(\frac{-15}{20}\)[/tex] and [tex]\(\frac{21}{-27}\)[/tex]
Here,
- [tex]\(a = -15\)[/tex]
- [tex]\(b = 20\)[/tex]
- [tex]\(c = 21\)[/tex]
- [tex]\(d = -27\)[/tex]
Now, calculate [tex]\(a \cdot d\)[/tex] and [tex]\(b \cdot c\)[/tex]:
[tex]\[ (-15) \cdot (-27) = 405 \][/tex]
[tex]\[ 20 \cdot 21 = 420 \][/tex]
Since [tex]\(405 \neq 420\)[/tex], these fractions are not equal.
### (c) [tex]\(\frac{-8}{24}\)[/tex] and [tex]\(\frac{7}{-21}\)[/tex]
Given fractions: [tex]\(\frac{-8}{24}\)[/tex] and [tex]\(\frac{7}{-21}\)[/tex]
Here,
- [tex]\(a = -8\)[/tex]
- [tex]\(b = 24\)[/tex]
- [tex]\(c = 7\)[/tex]
- [tex]\(d = -21\)[/tex]
Now, calculate [tex]\(a \cdot d\)[/tex] and [tex]\(b \cdot c\)[/tex]:
[tex]\[ (-8) \cdot (-21) = 168 \][/tex]
[tex]\[ 24 \cdot 7 = 168 \][/tex]
Since [tex]\(168 = 168\)[/tex], these fractions are equal.
### (d) [tex]\(\frac{-6}{-10}\)[/tex] and [tex]\(\frac{11}{15}\)[/tex]
Given fractions: [tex]\(\frac{-6}{-10}\)[/tex] and [tex]\(\frac{11}{15}\)[/tex]
Here,
- [tex]\(a = -6\)[/tex]
- [tex]\(b = -10\)[/tex]
- [tex]\(c = 11\)[/tex]
- [tex]\(d = 15\)[/tex]
Now, calculate [tex]\(a \cdot d\)[/tex] and [tex]\(b \cdot c\)[/tex]:
[tex]\[ (-6) \cdot 15 = -90 \][/tex]
[tex]\[ (-10) \cdot 11 = -110 \][/tex]
Since [tex]\(-90 \neq -110\)[/tex], these fractions are not equal.
### Conclusion
Only one pair, pair (c), is equal:
- [tex]\(\frac{-8}{24}\)[/tex] and [tex]\(\frac{7}{-21}\)[/tex]
So, the answer is:
(c) [tex]\(\frac{-8}{24}\)[/tex] and [tex]\(\frac{7}{-21}\)[/tex].
