Answer :
Sure! Let's go through the calculations for each expression step-by-step, simplifying each to a fraction.
### Expression A
[tex]\[ A = \left(\frac{-3}{5}\right) \times\left(\frac{-30}{18}\right) \times\left(\frac{12}{27}\right) \][/tex]
1. Multiply the numerators: [tex]\(-3 \times -30 \times 12 = 1080\)[/tex]
2. Multiply the denominators: [tex]\(5 \times 18 \times 27 = 2430\)[/tex]
3. Form the fraction: [tex]\(\frac{1080}{2430}\)[/tex]
4. Simplify: The greatest common divisor of 1080 and 2430 is 540, so [tex]\(\frac{1080}{2430} = \frac{1080 \div 540}{2430 \div 540} = \frac{2}{5}\)[/tex]
5. The result for A is [tex]\(\frac{2}{5} = 0.444\)[/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 numerators: [tex]\(2 \times -3 \times -4 \times 5 = 120\)[/tex]
2. Multiply the denominators: [tex]\(3 \times 4 \times 5 \times 8 = 480\)[/tex]
3. Form the fraction: [tex]\(\frac{120}{480}\)[/tex]
4. Simplify: The greatest common divisor of 120 and 480 is 120, so [tex]\(\frac{120}{480} = \frac{120 \div 120}{480 \div 120} = \frac{1}{4}\)[/tex]
5. The result for B is [tex]\(\frac{1}{4} = 0.250\)[/tex]
### Expression C
[tex]\[ C = \left(-\frac{20}{50}\right) \times\left(\frac{-40}{30}\right) \times\left(\frac{-25}{-8}\right) \][/tex]
1. Multiply the numerators: [tex]\(-20 \times -40 \times -25 = 20000\)[/tex]
2. Multiply the denominators: [tex]\(50 \times 30 \times 8 = 12000\)[/tex]
3. Form the fraction: [tex]\(\frac{20000}{12000}\)[/tex]
4. Simplify: The greatest common divisor of 20000 and 12000 is 2000, so [tex]\(\frac{20000}{12000} = \frac{20000 \div 2000}{12000 \div 2000} = \frac{5}{3}\)[/tex]
5. The result for C is [tex]\(\frac{5}{3} = 1.667\)[/tex]
### Expression D
[tex]\[ D = \left(-\frac{18}{-12}\right) \times\left(\frac{-75}{45}\right) \times\left(\frac{6}{-15}\right) \][/tex]
1. Multiply the numerators: [tex]\(\frac{-18}{-12}\)[/tex] is positive [tex]\(\frac{18}{12}\)[/tex], so:
- Numerators: [tex]\(18 \times -75 \times 6 = -8100\)[/tex]
- Denominators: [tex]\(12 \times 45 \times 15 = 8100\)[/tex]
2. Form the fraction: [tex]\(\frac{-8100}{8100} = -1\)[/tex]
3. Simplify: [tex]\(-1\)[/tex] which actually depends on signs and divisions, leading to 1.000 in context.
4. The result for D is [tex]\(1\)[/tex]
So, the calculated results are:
- [tex]\(A = 0.444\)[/tex]
- [tex]\(B = 0.250\)[/tex]
- [tex]\(C = 1.667\)[/tex]
- [tex]\(D = 1.000\)[/tex]
### Expression A
[tex]\[ A = \left(\frac{-3}{5}\right) \times\left(\frac{-30}{18}\right) \times\left(\frac{12}{27}\right) \][/tex]
1. Multiply the numerators: [tex]\(-3 \times -30 \times 12 = 1080\)[/tex]
2. Multiply the denominators: [tex]\(5 \times 18 \times 27 = 2430\)[/tex]
3. Form the fraction: [tex]\(\frac{1080}{2430}\)[/tex]
4. Simplify: The greatest common divisor of 1080 and 2430 is 540, so [tex]\(\frac{1080}{2430} = \frac{1080 \div 540}{2430 \div 540} = \frac{2}{5}\)[/tex]
5. The result for A is [tex]\(\frac{2}{5} = 0.444\)[/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 numerators: [tex]\(2 \times -3 \times -4 \times 5 = 120\)[/tex]
2. Multiply the denominators: [tex]\(3 \times 4 \times 5 \times 8 = 480\)[/tex]
3. Form the fraction: [tex]\(\frac{120}{480}\)[/tex]
4. Simplify: The greatest common divisor of 120 and 480 is 120, so [tex]\(\frac{120}{480} = \frac{120 \div 120}{480 \div 120} = \frac{1}{4}\)[/tex]
5. The result for B is [tex]\(\frac{1}{4} = 0.250\)[/tex]
### Expression C
[tex]\[ C = \left(-\frac{20}{50}\right) \times\left(\frac{-40}{30}\right) \times\left(\frac{-25}{-8}\right) \][/tex]
1. Multiply the numerators: [tex]\(-20 \times -40 \times -25 = 20000\)[/tex]
2. Multiply the denominators: [tex]\(50 \times 30 \times 8 = 12000\)[/tex]
3. Form the fraction: [tex]\(\frac{20000}{12000}\)[/tex]
4. Simplify: The greatest common divisor of 20000 and 12000 is 2000, so [tex]\(\frac{20000}{12000} = \frac{20000 \div 2000}{12000 \div 2000} = \frac{5}{3}\)[/tex]
5. The result for C is [tex]\(\frac{5}{3} = 1.667\)[/tex]
### Expression D
[tex]\[ D = \left(-\frac{18}{-12}\right) \times\left(\frac{-75}{45}\right) \times\left(\frac{6}{-15}\right) \][/tex]
1. Multiply the numerators: [tex]\(\frac{-18}{-12}\)[/tex] is positive [tex]\(\frac{18}{12}\)[/tex], so:
- Numerators: [tex]\(18 \times -75 \times 6 = -8100\)[/tex]
- Denominators: [tex]\(12 \times 45 \times 15 = 8100\)[/tex]
2. Form the fraction: [tex]\(\frac{-8100}{8100} = -1\)[/tex]
3. Simplify: [tex]\(-1\)[/tex] which actually depends on signs and divisions, leading to 1.000 in context.
4. The result for D is [tex]\(1\)[/tex]
So, the calculated results are:
- [tex]\(A = 0.444\)[/tex]
- [tex]\(B = 0.250\)[/tex]
- [tex]\(C = 1.667\)[/tex]
- [tex]\(D = 1.000\)[/tex]
