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Solve the following: 1) 5 3/4 - 7(2 1/2 + 1 1/3); 2) Find three consecutive numbers that add up to 87.

Mathematics

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9)

  1. Convert mixed numbers to improper fractions.

    534=5×4+34=2345\frac{3}{4} = \frac{5 \times 4 + 3}{4} = \frac{23}{4} 212=2×2+12=522\frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2} 113=1×3+13=431\frac{1}{3} = \frac{1 \times 3 + 1}{3} = \frac{4}{3}
  2. Solve the expression inside the parentheses.

    52+43=5×32×3+4×23×2=156+86=15+86=236\frac{5}{2} + \frac{4}{3} = \frac{5 \times 3}{2 \times 3} + \frac{4 \times 2}{3 \times 2} = \frac{15}{6} + \frac{8}{6} = \frac{15+8}{6} = \frac{23}{6}
  3. Multiply the result by 7.

    7×236=16167 \times \frac{23}{6} = \frac{161}{6}
  4. Perform the subtraction.

    234−1616\frac{23}{4} - \frac{161}{6}

    Find a common denominator, which is 12.

    23×34×3−161×26×2=6912−32212=69−32212=−25312\frac{23 \times 3}{4 \times 3} - \frac{161 \times 2}{6 \times 2} = \frac{69}{12} - \frac{322}{12} = \frac{69 - 322}{12} = \frac{-253}{12}

    The answer can also be expressed as a mixed number:

    −25312=−21112\frac{-253}{12} = -21\frac{1}{12}

    The final answer is −25312\boxed{-\frac{253}{12}}.

    10)

  5. Define the three consecutive numbers.

    Let the first number be xx.
    The second number is x+1x+1.
    The third number is x+2x+2.

  6. Set up the equation based on their sum.

    x+(x+1)+(x+2)=87x + (x+1) + (x+2) = 87
  7. Solve for xx.

    3x+3=873x + 3 = 87 3x=87−33x = 87 - 3 3x=843x = 84 x=843x = \frac{84}{3} x=28x = 28
  8. Find the three numbers.

    First number: 28
    Second number: 28+1=2928+1 = 29
    Third number: 28+2=3028+2 = 30
    The final answer is 28, 29, and 30.

    11)

  9. Write the decimal as a fraction.

    0.312=31210000.312 = \frac{312}{1000}
  10. Simplify the fraction by dividing the numerator and denominator by their greatest common divisor.

    Divide by 2:

    312÷21000÷2=156500\frac{312 \div 2}{1000 \div 2} = \frac{156}{500}

    Divide by 2 again:

    156÷2500÷2=78250\frac{156 \div 2}{500 \div 2} = \frac{78}{250}

    Divide by 2 again:

    78÷2250÷2=39125\frac{78 \div 2}{250 \div 2} = \frac{39}{125}

    The final answer is 39125\boxed{\frac{39}{125}}.

    14)

  11. Perform division.

    12.8÷0.8=1612.8 \div 0.8 = 16
  12. Perform multiplication.

    0.2×3=0.60.2 \times 3 = 0.6
  13. Substitute these values back into the expression and perform subtraction and addition from left to right.

    16−0.6+1.216 - 0.6 + 1.2 15.4+1.215.4 + 1.2 16.616.6

    The final answer is 16.6\boxed{16.6}.

    15)

    a) Determine the fraction that represents the cost of each bunch of vegetables.

  14. Find the total number of ratio parts.

    The ratio of carrots : spinach : onions is 2 : 4 : 3.
    Total parts = 2+4+3=92 + 4 + 3 = 9.

  15. Determine the fraction of the total cost for each type of vegetable.

    Fraction for carrots = Carrots ratio partTotal ratio parts=29\frac{\text{Carrots ratio part}}{\text{Total ratio parts}} = \frac{2}{9}
    Fraction for spinach = Spinach ratio partTotal ratio parts=49\frac{\text{Spinach ratio part}}{\text{Total ratio parts}} = \frac{4}{9}
    Fraction for onions = Onions ratio partTotal ratio parts=39\frac{\text{Onions ratio part}}{\text{Total ratio parts}} = \frac{3}{9}

    The fractions representing the cost of each bunch of vegetables are:

    • Carrots: 29\boxed{\frac{2}{9}}
    • Spinach: 49\boxed{\frac{4}{9}}
    • Onions: 39\boxed{\frac{3}{9}}
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