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Simplify the following expressions: a) (2 1/2 + 1 1/3) / (1 2/3 - 1/2) b) (0.25 0.35) / (0.14 0.05)

Mathematics

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1a) i)

  1. Convert mixed numbers to improper fractions.

    212=522\frac{1}{2} = \frac{5}{2} 113=431\frac{1}{3} = \frac{4}{3} 123=531\frac{2}{3} = \frac{5}{3}

    The expression becomes:

    (52+43)÷(53−12)\left(\frac{5}{2} + \frac{4}{3}\right) \div \left(\frac{5}{3} - \frac{1}{2}\right)

  2. Perform the addition inside the first parenthesis.

    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. Perform the subtraction inside the second parenthesis.

    53−12=5×23×2−1×32×3=106−36=10−36=76\frac{5}{3} - \frac{1}{2} = \frac{5 \times 2}{3 \times 2} - \frac{1 \times 3}{2 \times 3} = \frac{10}{6} - \frac{3}{6} = \frac{10-3}{6} = \frac{7}{6}

  4. Perform the division.

    236÷76=236×67=237\frac{23}{6} \div \frac{7}{6} = \frac{23}{6} \times \frac{6}{7} = \frac{23}{7}

    The simplified value is 237\boxed{\frac{23}{7}}.

    1a) ii)

  5. Rewrite the decimals as fractions or powers of 10.

    0.25=25×10−20.25 = 25 \times 10^{-2} 0.35=35×10−20.35 = 35 \times 10^{-2} 0.14=14×10−20.14 = 14 \times 10^{-2} 0.05=5×10−20.05 = 5 \times 10^{-2}

  6. Substitute these into the expression.

    0.25×0.350.14×0.05=(25×10−2)×(35×10−2)(14×10−2)×(5×10−2)\frac{0.25 \times 0.35}{0.14 \times 0.05} = \frac{(25 \times 10^{-2}) \times (35 \times 10^{-2})}{(14 \times 10^{-2}) \times (5 \times 10^{-2})}

  7. Simplify by cancelling common factors.

    =25×35×10−414×5×10−4= \frac{25 \times 35 \times 10^{-4}}{14 \times 5 \times 10^{-4}} =25×3514×5= \frac{25 \times 35}{14 \times 5} =(5×5)×(5×7)(2×7)×5= \frac{(5 \times 5) \times (5 \times 7)}{(2 \times 7) \times 5} =5×5×5×72×7×5= \frac{5 \times 5 \times 5 \times 7}{2 \times 7 \times 5}

    Cancel one 5 and one 7 from numerator and denominator:

    =5×52= \frac{5 \times 5}{2} =252= \frac{25}{2}

    The evaluated value is 252\boxed{\frac{25}{2}} or 12.5\boxed{12.5}.

    1b)

  8. Define the shares using a variable.

    Let Akon's share be xx.

  9. Express Atango's share in terms of xx.

    Atango takes two more than Akon, so Atango's share is x+2x + 2.

  10. Express Abomo's share in terms of xx.

    Abomo takes twice as much as Atango, so Abomo's share is 2×(x+2)2 \times (x+2).

  11. Express their shares as a ratio.

    Akon : Atango : Abomo = x:(x+2):2(x+2)x : (x+2) : 2(x+2)
    The ratio of their shares is x:(x+2):2(x+2)\boxed{x : (x+2) : 2(x+2)}.

    2a)
    Prime numbers are numbers greater than 1 that are only divisible by 1 and themselves.
    The prime numbers between 4 and 15 are 5, 7, 11, 13.
    The set is {5,7,11,13}\boxed{\{5, 7, 11, 13\}}.

    2b)
    Square numbers are the result of multiplying an integer by itself.
    12=11^2 = 1
    22=42^2 = 4
    32=93^2 = 9
    42=164^2 = 16
    52=255^2 = 25 (which is not less than 20)
    The square numbers less than 20 are 1, 4, 9, 16.
    The set is {1,4,9,16}\boxed{\{1, 4, 9, 16\}}.

    2c)
    Multiples of 3 are numbers that can be divided by 3 without a remainder.
    The multiples of 3 between 11 and 22 are 12, 15, 18, 21.
    The set is {12,15,18,21}\boxed{\{12, 15, 18, 21\}}.

    2d)
    Even numbers are integers that are divisible by 2.
    The even numbers between 10 and 22 (exclusive of 10 and 22) are 12, 14, 16, 18, 20.
    The set is $\boxed{{12,

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