Answer :
Sure! Let's calculate each expression step-by-step and express them as simplified fractions.
### Expression A
[tex]\[ A = \left(\frac{-3}{5}\right) \times \left(\frac{-30}{18}\right) \times \left(\frac{12}{27}\right) \][/tex]
1. First fraction: [tex]\(\frac{-3}{5}\)[/tex]
2. Second fraction: [tex]\(\frac{-30}{18}\)[/tex] can be simplified. Both numerator and denominator can be divided by 6:
[tex]\[\frac{-30}{18} = \frac{-5}{3}\][/tex]
3. Third fraction: [tex]\(\frac{12}{27}\)[/tex] can be simplified. Both numerator and denominator can be divided by 3:
[tex]\[\frac{12}{27} = \frac{4}{9}\][/tex]
4. Multiply the simplified fractions:
[tex]\[\left(\frac{-3}{5}\right) \times \left(\frac{-5}{3}\right) \times \left(\frac{4}{9}\right)\][/tex]
5. Compute:
[tex]\[\frac{-3 \times -5 \times 4}{5 \times 3 \times 9} = \frac{60}{135}\][/tex]
6. Simplify [tex]\(\frac{60}{135}\)[/tex] by dividing both the numerator and the denominator by 15:
[tex]\[\frac{60}{135} = \frac{4}{9}\][/tex]
So, the result for expression [tex]\(A\)[/tex] is [tex]\(\frac{4}{9}\)[/tex].
### Expression B
[tex]\[ B = \frac{2}{3} \times \left(\frac{-3}{4}\right) \times \left(\frac{-4}{5}\right) \times \frac{5}{8} \][/tex]
1. Multiply the fractions:
[tex]\[\frac{2}{3} \times \frac{-3}{4} \times \frac{-4}{5} \times \frac{5}{8}\][/tex]
2. Compute:
[tex]\[\frac{2 \times -3 \times -4 \times 5}{3 \times 4 \times 5 \times 8} = \frac{120}{480}\][/tex]
3. Simplify [tex]\(\frac{120}{480}\)[/tex] by dividing both the numerator and the denominator by 120:
[tex]\[\frac{120}{480} = \frac{1}{4}\][/tex]
So, the result for expression [tex]\(B\)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
### Expression C
[tex]\[ C = \left(-\frac{20}{50}\right) \times \left(\frac{-40}{30}\right) \times \left(\frac{-25}{-8}\right) \][/tex]
1. First fraction: [tex]\(\frac{-20}{50}\)[/tex] can be simplified. Both numerator and denominator can be divided by 10:
[tex]\[\frac{-20}{50} = \frac{-2}{5}\][/tex]
2. Second fraction: [tex]\(\frac{-40}{30}\)[/tex] can be simplified. Both numerator and denominator can be divided by 10:
[tex]\[\frac{-40}{30} = \frac{-4}{3}\][/tex]
3. Third fraction: [tex]\(\frac{-25}{-8}\)[/tex] simplifies to [tex]\(\frac{25}{8}\)[/tex] since both signs cancel each other.
4. Multiply the simplified fractions:
[tex]\[\left(\frac{-2}{5}\right) \times \left(\frac{-4}{3}\right) \times \left(\frac{25}{8}\right)\][/tex]
5. Compute:
[tex]\[\frac{-2 \times -4 \times 25}{5 \times 3 \times 8} = \frac{200}{120}\][/tex]
6. Simplify [tex]\(\frac{200}{120}\)[/tex] by dividing both the numerator and the denominator by 40:
[tex]\[\frac{200}{120} = \frac{5}{3}\][/tex]
So, the result for expression [tex]\(C\)[/tex] is [tex]\(\frac{5}{3}\)[/tex].
### Expression D
[tex]\[ D = \left(-\frac{18}{-12}\right) \times \left(\frac{-75}{45}\right) \times \left(\frac{6}{-15}\right) \][/tex]
1. First fraction: [tex]\(\frac{-18}{-12}\)[/tex] simplifies to [tex]\(\frac{18}{12}\)[/tex] as the signs cancel. Then simplify further by dividing both by 6:
[tex]\[\frac{18}{12} = \frac{3}{2}\][/tex]
2. Second fraction: [tex]\(\frac{-75}{45}\)[/tex] can be simplified. Both numerator and denominator can be divided by 15:
[tex]\[\frac{-75}{45} = \frac{-5}{3}\][/tex]
3. Third fraction: [tex]\(\frac{6}{-15}\)[/tex] can be simplified. Both numerator and denominator can be divided by 3:
[tex]\[\frac{6}{-15} = \frac{-2}{5}\][/tex]
4. Multiply the simplified fractions:
[tex]\[\left(\frac{3}{2}\right) \times \left(\frac{-5}{3}\right) \times \left(\frac{-2}{5}\right)\][/tex]
5. Compute:
[tex]\[\frac{3 \times -5 \times -2}{2 \times 3 \times 5} = \frac{30}{30} = 1\][/tex]
So, the result for expression [tex]\(D\)[/tex] is [tex]\(1\)[/tex].
In summary:
- [tex]\( A = \frac{4}{9} \)[/tex]
- [tex]\( B = \frac{1}{4} \)[/tex]
- [tex]\( C = \frac{5}{3} \)[/tex]
- [tex]\( D = 1 \)[/tex]
### Expression A
[tex]\[ A = \left(\frac{-3}{5}\right) \times \left(\frac{-30}{18}\right) \times \left(\frac{12}{27}\right) \][/tex]
1. First fraction: [tex]\(\frac{-3}{5}\)[/tex]
2. Second fraction: [tex]\(\frac{-30}{18}\)[/tex] can be simplified. Both numerator and denominator can be divided by 6:
[tex]\[\frac{-30}{18} = \frac{-5}{3}\][/tex]
3. Third fraction: [tex]\(\frac{12}{27}\)[/tex] can be simplified. Both numerator and denominator can be divided by 3:
[tex]\[\frac{12}{27} = \frac{4}{9}\][/tex]
4. Multiply the simplified fractions:
[tex]\[\left(\frac{-3}{5}\right) \times \left(\frac{-5}{3}\right) \times \left(\frac{4}{9}\right)\][/tex]
5. Compute:
[tex]\[\frac{-3 \times -5 \times 4}{5 \times 3 \times 9} = \frac{60}{135}\][/tex]
6. Simplify [tex]\(\frac{60}{135}\)[/tex] by dividing both the numerator and the denominator by 15:
[tex]\[\frac{60}{135} = \frac{4}{9}\][/tex]
So, the result for expression [tex]\(A\)[/tex] is [tex]\(\frac{4}{9}\)[/tex].
### Expression B
[tex]\[ B = \frac{2}{3} \times \left(\frac{-3}{4}\right) \times \left(\frac{-4}{5}\right) \times \frac{5}{8} \][/tex]
1. Multiply the fractions:
[tex]\[\frac{2}{3} \times \frac{-3}{4} \times \frac{-4}{5} \times \frac{5}{8}\][/tex]
2. Compute:
[tex]\[\frac{2 \times -3 \times -4 \times 5}{3 \times 4 \times 5 \times 8} = \frac{120}{480}\][/tex]
3. Simplify [tex]\(\frac{120}{480}\)[/tex] by dividing both the numerator and the denominator by 120:
[tex]\[\frac{120}{480} = \frac{1}{4}\][/tex]
So, the result for expression [tex]\(B\)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
### Expression C
[tex]\[ C = \left(-\frac{20}{50}\right) \times \left(\frac{-40}{30}\right) \times \left(\frac{-25}{-8}\right) \][/tex]
1. First fraction: [tex]\(\frac{-20}{50}\)[/tex] can be simplified. Both numerator and denominator can be divided by 10:
[tex]\[\frac{-20}{50} = \frac{-2}{5}\][/tex]
2. Second fraction: [tex]\(\frac{-40}{30}\)[/tex] can be simplified. Both numerator and denominator can be divided by 10:
[tex]\[\frac{-40}{30} = \frac{-4}{3}\][/tex]
3. Third fraction: [tex]\(\frac{-25}{-8}\)[/tex] simplifies to [tex]\(\frac{25}{8}\)[/tex] since both signs cancel each other.
4. Multiply the simplified fractions:
[tex]\[\left(\frac{-2}{5}\right) \times \left(\frac{-4}{3}\right) \times \left(\frac{25}{8}\right)\][/tex]
5. Compute:
[tex]\[\frac{-2 \times -4 \times 25}{5 \times 3 \times 8} = \frac{200}{120}\][/tex]
6. Simplify [tex]\(\frac{200}{120}\)[/tex] by dividing both the numerator and the denominator by 40:
[tex]\[\frac{200}{120} = \frac{5}{3}\][/tex]
So, the result for expression [tex]\(C\)[/tex] is [tex]\(\frac{5}{3}\)[/tex].
### Expression D
[tex]\[ D = \left(-\frac{18}{-12}\right) \times \left(\frac{-75}{45}\right) \times \left(\frac{6}{-15}\right) \][/tex]
1. First fraction: [tex]\(\frac{-18}{-12}\)[/tex] simplifies to [tex]\(\frac{18}{12}\)[/tex] as the signs cancel. Then simplify further by dividing both by 6:
[tex]\[\frac{18}{12} = \frac{3}{2}\][/tex]
2. Second fraction: [tex]\(\frac{-75}{45}\)[/tex] can be simplified. Both numerator and denominator can be divided by 15:
[tex]\[\frac{-75}{45} = \frac{-5}{3}\][/tex]
3. Third fraction: [tex]\(\frac{6}{-15}\)[/tex] can be simplified. Both numerator and denominator can be divided by 3:
[tex]\[\frac{6}{-15} = \frac{-2}{5}\][/tex]
4. Multiply the simplified fractions:
[tex]\[\left(\frac{3}{2}\right) \times \left(\frac{-5}{3}\right) \times \left(\frac{-2}{5}\right)\][/tex]
5. Compute:
[tex]\[\frac{3 \times -5 \times -2}{2 \times 3 \times 5} = \frac{30}{30} = 1\][/tex]
So, the result for expression [tex]\(D\)[/tex] is [tex]\(1\)[/tex].
In summary:
- [tex]\( A = \frac{4}{9} \)[/tex]
- [tex]\( B = \frac{1}{4} \)[/tex]
- [tex]\( C = \frac{5}{3} \)[/tex]
- [tex]\( D = 1 \)[/tex]
