Answer :
Let's simplify each expression step by step to express the result as a simpler fraction.
### Expression A:
[tex]\[ A = \left(\frac{-3}{5}\right) \times \left(\frac{-30}{18}\right) \times \left(\frac{12}{27}\right) \][/tex]
1. Simplify [tex]\(\frac{-30}{18}\)[/tex]:
[tex]\(-30\)[/tex] and [tex]\(18\)[/tex] are both divisible by [tex]\(6\)[/tex].
[tex]\[ \frac{-30}{18} = \frac{-30 \div 6}{18 \div 6} = \frac{-5}{3} \][/tex]
2. Simplify [tex]\(\frac{12}{27}\)[/tex]:
[tex]\(12\)[/tex] and [tex]\(27\)[/tex] are both divisible by [tex]\(3\)[/tex].
[tex]\[ \frac{12}{27} = \frac{12 \div 3}{27 \div 3} = \frac{4}{9} \][/tex]
3. Multiply the fractions:
[tex]\[ \left(\frac{-3}{5}\right) \times \left(\frac{-5}{3}\right) \times \left(\frac{4}{9}\right) \][/tex]
[tex]\[ \frac{-3 \times -5 \times 4}{5 \times 3 \times 9} = \frac{60}{135} \][/tex]
4. Simplify [tex]\(\frac{60}{135}\)[/tex]:
[tex]\(60\)[/tex] and [tex]\(135\)[/tex] are both divisible by [tex]\(15\)[/tex].
[tex]\[ \frac{60}{135} = \frac{60 \div 15}{135 \div 15} = \frac{4}{9} \][/tex]
So, the simplified result for [tex]\(A\)[/tex] is [tex]\(\frac{4}{9}\)[/tex].
### Expression B:
[tex]\[ B = \left(\frac{2}{3}\right) \times \left(\frac{-3}{4}\right) \times \left(\frac{-4}{5}\right) \times \frac{5}{8} \][/tex]
1. Multiply the fractions:
[tex]\[ \frac{2 \times -3 \times -4 \times 5}{3 \times 4 \times 5 \times 8} \][/tex]
2. Cancelling out:
[tex]\[ \frac{2 \times (-3) \times (-4) \times 5}{3 \times 4 \times 5 \times 8} \][/tex]
3. Multiplying the numerators and denominators:
[tex]\[ \frac{120}{480} \][/tex]
4. Simplify [tex]\(\frac{120}{480}\)[/tex]:
[tex]\(120\)[/tex] and [tex]\(480\)[/tex] are both divisible by [tex]\(120\)[/tex].
[tex]\[ \frac{120}{480} = \frac{120 \div 120}{480 \div 120} = \frac{1}{4} \][/tex]
So, the simplified result for [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. Simplify [tex]\(\frac{-20}{50}\)[/tex]:
[tex]\(-20\)[/tex] and [tex]\(50\)[/tex] are both divisible by [tex]\(10\)[/tex].
[tex]\[ \frac{-20}{50} = \frac{-2}{5} \][/tex]
2. Simplify [tex]\(\frac{-40}{30}\)[/tex]:
[tex]\(-40\)[/tex] and [tex]\(30\)[/tex] are both divisible by [tex]\(10\)[/tex].
[tex]\[ \frac{-40}{30} = \frac{-4}{3} \][/tex]
3. Simplify [tex]\(\frac{-25}{-8}\)[/tex]:
[tex]\(\frac{-25}{-8} = \frac{25}{8}\)[/tex] since two negatives make a positive.
4. Multiply the fractions:
[tex]\[ \frac{-2 \times -4 \times 25}{5 \times 3 \times 8} \][/tex]
5. Multiplying the numerators and denominators:
[tex]\[ \frac{200}{120} \][/tex]
6. Simplify [tex]\(\frac{200}{120}\)[/tex]:
[tex]\(200\)[/tex] and [tex]\(120\)[/tex] are both divisible by [tex]\(40\)[/tex].
[tex]\[ \frac{200}{120} = \frac{200 \div 40}{120 \div 40} = \frac{5}{3} \][/tex]
So, the simplified result for [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. Simplify [tex]\(\frac{-18}{-12}\)[/tex]:
[tex]\(-18\)[/tex] and [tex]\(-12\)[/tex] simplifies to [tex]\(\frac{3}{2}\)[/tex].
2. Simplify [tex]\(\frac{-75}{45}\)[/tex]:
[tex]\(-75\)[/tex] and [tex]\(45\)[/tex] are both divisible by [tex]\(15\)[/tex].
[tex]\[ \frac{-75}{45} = \frac{-5}{3} \][/tex]
3. Simplify [tex]\(\frac{6}{-15}\)[/tex]:
[tex]\(6\)[/tex] and [tex]\(-15\)[/tex] are both divisible by [tex]\(3\)[/tex].
[tex]\[ \frac{6}{-15} = \frac{-2}{5} \][/tex]
4. Multiply the fractions:
[tex]\[ \frac{3 \times -5 \times -2}{2 \times 3 \times 5} \][/tex]
5. Multiplying the numerators and denominators:
[tex]\[ \frac{30}{30} \][/tex]
6. [tex]\(\frac{30}{30} = 1\)[/tex].
So, the simplified result for [tex]\(D\)[/tex] is [tex]\(1\)[/tex].
To summarize:
- [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. Simplify [tex]\(\frac{-30}{18}\)[/tex]:
[tex]\(-30\)[/tex] and [tex]\(18\)[/tex] are both divisible by [tex]\(6\)[/tex].
[tex]\[ \frac{-30}{18} = \frac{-30 \div 6}{18 \div 6} = \frac{-5}{3} \][/tex]
2. Simplify [tex]\(\frac{12}{27}\)[/tex]:
[tex]\(12\)[/tex] and [tex]\(27\)[/tex] are both divisible by [tex]\(3\)[/tex].
[tex]\[ \frac{12}{27} = \frac{12 \div 3}{27 \div 3} = \frac{4}{9} \][/tex]
3. Multiply the fractions:
[tex]\[ \left(\frac{-3}{5}\right) \times \left(\frac{-5}{3}\right) \times \left(\frac{4}{9}\right) \][/tex]
[tex]\[ \frac{-3 \times -5 \times 4}{5 \times 3 \times 9} = \frac{60}{135} \][/tex]
4. Simplify [tex]\(\frac{60}{135}\)[/tex]:
[tex]\(60\)[/tex] and [tex]\(135\)[/tex] are both divisible by [tex]\(15\)[/tex].
[tex]\[ \frac{60}{135} = \frac{60 \div 15}{135 \div 15} = \frac{4}{9} \][/tex]
So, the simplified result for [tex]\(A\)[/tex] is [tex]\(\frac{4}{9}\)[/tex].
### Expression B:
[tex]\[ B = \left(\frac{2}{3}\right) \times \left(\frac{-3}{4}\right) \times \left(\frac{-4}{5}\right) \times \frac{5}{8} \][/tex]
1. Multiply the fractions:
[tex]\[ \frac{2 \times -3 \times -4 \times 5}{3 \times 4 \times 5 \times 8} \][/tex]
2. Cancelling out:
[tex]\[ \frac{2 \times (-3) \times (-4) \times 5}{3 \times 4 \times 5 \times 8} \][/tex]
3. Multiplying the numerators and denominators:
[tex]\[ \frac{120}{480} \][/tex]
4. Simplify [tex]\(\frac{120}{480}\)[/tex]:
[tex]\(120\)[/tex] and [tex]\(480\)[/tex] are both divisible by [tex]\(120\)[/tex].
[tex]\[ \frac{120}{480} = \frac{120 \div 120}{480 \div 120} = \frac{1}{4} \][/tex]
So, the simplified result for [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. Simplify [tex]\(\frac{-20}{50}\)[/tex]:
[tex]\(-20\)[/tex] and [tex]\(50\)[/tex] are both divisible by [tex]\(10\)[/tex].
[tex]\[ \frac{-20}{50} = \frac{-2}{5} \][/tex]
2. Simplify [tex]\(\frac{-40}{30}\)[/tex]:
[tex]\(-40\)[/tex] and [tex]\(30\)[/tex] are both divisible by [tex]\(10\)[/tex].
[tex]\[ \frac{-40}{30} = \frac{-4}{3} \][/tex]
3. Simplify [tex]\(\frac{-25}{-8}\)[/tex]:
[tex]\(\frac{-25}{-8} = \frac{25}{8}\)[/tex] since two negatives make a positive.
4. Multiply the fractions:
[tex]\[ \frac{-2 \times -4 \times 25}{5 \times 3 \times 8} \][/tex]
5. Multiplying the numerators and denominators:
[tex]\[ \frac{200}{120} \][/tex]
6. Simplify [tex]\(\frac{200}{120}\)[/tex]:
[tex]\(200\)[/tex] and [tex]\(120\)[/tex] are both divisible by [tex]\(40\)[/tex].
[tex]\[ \frac{200}{120} = \frac{200 \div 40}{120 \div 40} = \frac{5}{3} \][/tex]
So, the simplified result for [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. Simplify [tex]\(\frac{-18}{-12}\)[/tex]:
[tex]\(-18\)[/tex] and [tex]\(-12\)[/tex] simplifies to [tex]\(\frac{3}{2}\)[/tex].
2. Simplify [tex]\(\frac{-75}{45}\)[/tex]:
[tex]\(-75\)[/tex] and [tex]\(45\)[/tex] are both divisible by [tex]\(15\)[/tex].
[tex]\[ \frac{-75}{45} = \frac{-5}{3} \][/tex]
3. Simplify [tex]\(\frac{6}{-15}\)[/tex]:
[tex]\(6\)[/tex] and [tex]\(-15\)[/tex] are both divisible by [tex]\(3\)[/tex].
[tex]\[ \frac{6}{-15} = \frac{-2}{5} \][/tex]
4. Multiply the fractions:
[tex]\[ \frac{3 \times -5 \times -2}{2 \times 3 \times 5} \][/tex]
5. Multiplying the numerators and denominators:
[tex]\[ \frac{30}{30} \][/tex]
6. [tex]\(\frac{30}{30} = 1\)[/tex].
So, the simplified result for [tex]\(D\)[/tex] is [tex]\(1\)[/tex].
To summarize:
- [tex]\( A = \frac{4}{9} \)[/tex]
- [tex]\( B = \frac{1}{4} \)[/tex]
- [tex]\( C = \frac{5}{3} \)[/tex]
- [tex]\( D = 1 \)[/tex]
