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Calculate; i. 6 5 3 × 2 4 1 ÷ 3 2 1 + 5 2 1 ii. √ 8 5 × 4 3 ÷ (6)+2 4 The sum of K1050 is to be shared am 1 of the share of the first born. The th youngest gets of the share of the fir 3 6

Mathematics

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13 steps

Here are the solutions for Question 13 and 14:

13. Calculate:

Part i· 4 steps

635×214÷312+5126\frac{3}{5} \times 2\frac{1}{4} \div 3\frac{1}{2} + 5\frac{1}{2}

  1. Convert all mixed numbers to improper fractions.

    635=3356\frac{3}{5} = \frac{33}{5} 214=942\frac{1}{4} = \frac{9}{4} 312=723\frac{1}{2} = \frac{7}{2} 512=1125\frac{1}{2} = \frac{11}{2}

    The expression becomes:

    335×94÷72+112\frac{33}{5} \times \frac{9}{4} \div \frac{7}{2} + \frac{11}{2}
  2. Perform multiplication and division from left to right.

    First, multiply:

    335×94=29720\frac{33}{5} \times \frac{9}{4} = \frac{297}{20}

    The expression is now:

    29720÷72+112\frac{297}{20} \div \frac{7}{2} + \frac{11}{2}

    Next, divide (multiply by the reciprocal):

    29720×27=297×220×7=594140=29770\frac{297}{20} \times \frac{2}{7} = \frac{297 \times 2}{20 \times 7} = \frac{594}{140} = \frac{297}{70}

    The expression is now:

    29770+112\frac{297}{70} + \frac{11}{2}
  3. Perform addition.

    Find a common denominator, which is 70.

    112=11×352×35=38570\frac{11}{2} = \frac{11 \times 35}{2 \times 35} = \frac{385}{70} 29770+38570=297+38570=68270\frac{297}{70} + \frac{385}{70} = \frac{297 + 385}{70} = \frac{682}{70}
  4. Simplify the fraction.

    68270=34135\frac{682}{70} = \frac{341}{35}

    As a mixed number:

    926359\frac{26}{35}

    The calculated value is 34135 or 92635\boxed{\frac{341}{35} \text{ or } 9\frac{26}{35}}.

Part ii· 9 steps

58×34÷4(6)+2\sqrt{\frac{5}{8} \times \frac{3}{4} \div \frac{4}{(6)+2}}

  1. Simplify the denominator of the last fraction.

    (6)+2=8(6)+2 = 8

    The last fraction is 48\frac{4}{8}, which simplifies to 12\frac{1}{2}.
    The expression becomes:

    58×34÷12\sqrt{\frac{5}{8} \times \frac{3}{4} \div \frac{1}{2}}
  2. Perform multiplication and division inside the square root from left to right.

    First, multiply:

    58×34=1532\frac{5}{8} \times \frac{3}{4} = \frac{15}{32}

    The expression is now:

    1532÷12\sqrt{\frac{15}{32} \div \frac{1}{2}}

    Next, divide (multiply by the reciprocal):

    1532×21=3032=1516\frac{15}{32} \times \frac{2}{1} = \frac{30}{32} = \frac{15}{16}

    The expression is now:

    1516\sqrt{\frac{15}{16}}
  3. Calculate the square root.

    1516=1516=154\sqrt{\frac{15}{16}} = \frac{\sqrt{15}}{\sqrt{16}} = \frac{\sqrt{15}}{4}

    The calculated value is 154\boxed{\frac{\sqrt{15}}{4}}.

    14. The sum of K1050 is to be shared among three children. The second born gets 13\frac{1}{3} of the share of the first born. The youngest gets 16\frac{1}{6} of the share of the first born.

  4. Define variables for the shares.

    Let the share of the first born be FF.
    Let the share of the second born be SS.
    Let the share of the youngest born be YY.

  5. Express SS and YY in terms of FF.

    S=13FS = \frac{1}{3}F Y=16FY = \frac{1}{6}F
  6. Set up an equation for the total sum.

    The total sum is K1050, so:

    F+S+Y=1050F + S + Y = 1050

    Substitute the expressions for SS and YY:

    F+13F+16F=1050F + \frac{1}{3}F + \frac{1}{6}F = 1050
  7. Solve for FF.

    Find a common denominator for the fractions, which is 6:

    66F+26F+16F=1050\frac{6}{6}F + \frac{2}{6}F + \frac{1}{6}F = 1050

    Combine the terms:

    6+2+16F=1050\frac{6+2+1}{6}F = 1050 96F=1050\frac{9}{6}F = 1050

    Simplify the fraction 96\frac{9}{6} to 32\frac{3}{2}:

    32F=1050\frac{3}{2}F = 1050

    Multiply both sides by 23\frac{2}{3} to find FF:

    F=1050×23F = 1050 \times \frac{2}{3} F=21003F = \frac{2100}{3} F=700F = 700

    The first born's share is K700.

  8. Calculate the shares of the second and youngest born.

    Second born's share (SS):

    S=13F=13×700=7003≈K233.33S = \frac{1}{3}F = \frac{1}{3} \times 700 = \frac{700}{3} \approx \text{K}233.33

    Youngest born's share (YY):

    Y=16F=16×700=7006=3503≈K116.67Y = \frac{1}{6}F = \frac{1}{6} \times 700 = \frac{700}{6} = \frac{350}{3} \approx \text{K}116.67

    (Note: Shares are rounded to two decimal places for currency.)

  9. Verify the total sum.

    K700.00+K233.33+K116.67=K1050.00\text{K}700.00 + \text{K}233.33 + \text{K}116.67 = \text{K}1050.00

    The shares are:
    First born: K700.00\boxed{\text{K}700.00}
    Second born: K233.33\boxed{\text{K}233.33}
    Youngest born: K116.67\boxed{\text{K}116.67}

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