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Evaluate: (a) 2.4 (0.123 - 0.095) 0.009 / (0.015 * (0.042 + 0.030)); (b) Simplify: 1 3/4 + [5 - 2 3/4 / 3/5] - [ (3-1)/5 of 4 1/2 ]

Mathematics

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Answers

2 parts
a
0.56
b

1130\frac{11}{30}

Step-by-step solution

12 steps

Part a· 5 steps

Evaluate

2.4×(0.123−0.095)×0.0090.015×(0.042+0.030)\frac{2.4 \times (0.123 - 0.095) \times 0.009}{0.015 \times (0.042 + 0.030)}
  1. Simplify the expressions inside the parentheses.

    0.123−0.095=0.0280.123 - 0.095 = 0.028 0.042+0.030=0.0720.042 + 0.030 = 0.072
  2. Substitute these values back into the expression.

    2.4×0.028×0.0090.015×0.072\frac{2.4 \times 0.028 \times 0.009}{0.015 \times 0.072}
  3. Calculate the numerator.

    2.4×0.028=0.06722.4 \times 0.028 = 0.0672 0.0672×0.009=0.00060480.0672 \times 0.009 = 0.0006048
  4. Calculate the denominator.

    0.015×0.072=0.001080.015 \times 0.072 = 0.00108
  5. Divide the numerator by the denominator.

    0.00060480.00108=0.56\frac{0.0006048}{0.00108} = 0.56

    The evaluated value is 0.56\boxed{0.56}.

Part b

1130\frac{11}{30}

Simplify

134+[5−234÷35]−[3−15 of 412]1\frac{3}{4} + \left[5 - 2\frac{3}{4} \div \frac{3}{5}\right] - \left[\frac{3-1}{5} \text{ of } 4\frac{1}{2}\right]
  1. Convert all mixed numbers to improper fractions and simplify the first bracket in the third term.

    134=1×4+34=741\frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4} 234=2×4+34=1142\frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{11}{4} 412=4×2+12=924\frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{9}{2} 3−15=25\frac{3-1}{5} = \frac{2}{5}

    The expression becomes:

    74+[5−114÷35]−[25 of 92]\frac{7}{4} + \left[5 - \frac{11}{4} \div \frac{3}{5}\right] - \left[\frac{2}{5} \text{ of } \frac{9}{2}\right]
  2. Perform the division inside the first square bracket.

    114÷35=114×53=5512\frac{11}{4} \div \frac{3}{5} = \frac{11}{4} \times \frac{5}{3} = \frac{55}{12}

    The expression becomes:

    74+[5−5512]−[25 of 92]\frac{7}{4} + \left[5 - \frac{55}{12}\right] - \left[\frac{2}{5} \text{ of } \frac{9}{2}\right]
  3. Perform the subtraction inside the first square bracket.

    5−5512=5×1212−5512=6012−5512=5125 - \frac{55}{12} = \frac{5 \times 12}{12} - \frac{55}{12} = \frac{60}{12} - \frac{55}{12} = \frac{5}{12}

    The expression becomes:

    74+512−[25 of 92]\frac{7}{4} + \frac{5}{12} - \left[\frac{2}{5} \text{ of } \frac{9}{2}\right]
  4. Perform the "of" (multiplication) operation inside the second square bracket.

    25 of 92=25×92=1810=95\frac{2}{5} \text{ of } \frac{9}{2} = \frac{2}{5} \times \frac{9}{2} = \frac{18}{10} = \frac{9}{5}

    The expression becomes:

    74+512−95\frac{7}{4} + \frac{5}{12} - \frac{9}{5}
  5. Find a common denominator for 4, 12, and 5, which is 60.

    Convert the fractions.

    74=7×154×15=10560\frac{7}{4} = \frac{7 \times 15}{4 \times 15} = \frac{105}{60} 512=5×512×5=2560\frac{5}{12} = \frac{5 \times 5}{12 \times 5} = \frac{25}{60} 95=9×125×12=10860\frac{9}{5} = \frac{9 \times 12}{5 \times 12} = \frac{108}{60}

    The expression becomes:

    10560+2560−10860\frac{105}{60} + \frac{25}{60} - \frac{108}{60}
  6. Perform the addition and subtraction from left to right.

    105+25−10860=130−10860=2260\frac{105 + 25 - 108}{60} = \frac{130 - 108}{60} = \frac{22}{60}
  7. Simplify the fraction.

    2260=1130\frac{22}{60} = \frac{11}{30}

    The simplified value is 1130\boxed{\frac{11}{30}}.

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