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Evaluate, giving your answer as a fraction in its lowest terms: 3 1/2 - 1 3/4 + 2/3

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2 parts
a

2912\frac{29}{12}

b

3607 litres\frac{360}{7} \text{ litres}

Step-by-step solution

10 steps

Part a· 4 steps

To evaluate 312−134+233 \frac{1}{2} - 1 \frac{3}{4} + \frac{2}{3}

  1. Convert mixed numbers to improper fractions.

    312=3×2+12=723 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{7}{2} 134=1×4+34=741 \frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4}

    The expression becomes:

    72−74+23\frac{7}{2} - \frac{7}{4} + \frac{2}{3}

  2. Find a common denominator for 2, 4, and 3.

    The least common multiple (LCM) is 12.
    Convert each fraction to have a denominator of 12.

    72=7×62×6=4212\frac{7}{2} = \frac{7 \times 6}{2 \times 6} = \frac{42}{12} 74=7×34×3=2112\frac{7}{4} = \frac{7 \times 3}{4 \times 3} = \frac{21}{12} 23=2×43×4=812\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}

  3. Perform the subtraction and addition.

    4212−2112+812=42−21+812\frac{42}{12} - \frac{21}{12} + \frac{8}{12} = \frac{42 - 21 + 8}{12} =21+812= \frac{21 + 8}{12} =2912= \frac{29}{12}

  4. The fraction 2912\frac{29}{12} is in its lowest terms because 29 is a prime number and 12 is not a multiple of 29.

    The result can also be expressed as a mixed number: 25122 \frac{5}{12}.

    The evaluated expression is 2912\boxed{\frac{29}{12}}.

Part b

3607 litres\frac{360}{7} \text{ litres}

To find the capacity of the tank

  1. Let CC be the total capacity of the tank in litres.

    Initially, the tank is 25\frac{2}{5} full, so it contains 25C\frac{2}{5}C litres of water.
    After adding 18 litres, the tank becomes 34\frac{3}{4} full, meaning it contains 34C\frac{3}{4}C litres of water.

  2. Set up an equation based on the information.

    The difference between the final and initial amounts of water is 18 litres.

    34C−25C=18\frac{3}{4}C - \frac{2}{5}C = 18

  3. Find a common denominator for the fractions on the left side.

    The LCM of 4 and 5 is 20.

    3×54×5C−2×45×4C=18\frac{3 \times 5}{4 \times 5}C - \frac{2 \times 4}{5 \times 4}C = 18 1520C−820C=18\frac{15}{20}C - \frac{8}{20}C = 18

  4. Subtract the fractions.

    15−820C=18\frac{15 - 8}{20}C = 18 720C=18\frac{7}{20}C = 18

  5. Solve for CC by multiplying both sides by 207\frac{20}{7}.

    C=18×207C = 18 \times \frac{20}{7} C=3607C = \frac{360}{7}

  6. Convert the improper fraction to a mixed number or decimal if preferred, but the question asks for capacity, so a fraction is fine.

    C=5137 litresC = 51 \frac{3}{7} \text{ litres}

    The capacity of the tank is 3607 litres\boxed{\frac{360}{7} \text{ litres}}.

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