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Calculate the least share for problem 19c, and the HCF, LCM, and their product for problem 20a.

Mathematics

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Here are the solutions for problems 19c, 20a, 20b, and 20c.

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  1. Find the total number of parts in the ratio.

    Total parts=5+4+3=12\text{Total parts} = 5 + 4 + 3 = 12
  2. Calculate the value of one ratio part by dividing the total amount by the total parts.

    Value per part=₦240,00012=₦20,000\text{Value per part} = \frac{\text{₦}240,000}{12} = \text{₦}20,000
  3. The least share corresponds to the smallest number in the ratio, which is 3.

    Multiply the value per part by 3.

    Least share=3×₦20,000=₦60,000\text{Least share} = 3 \times \text{₦}20,000 = \text{₦}60,000

    The least share is ₦60,000\boxed{\text{₦}60,000}.

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  4. Find the prime factorization of each number

    - $9 = 3^2$
    - $12 = 2^2 \times 3$
    - $18 = 2 \times 3^2$
    
  5. Find the HCF (Highest Common Factor).

    This is the product of the lowest powers of common prime factors. The only common prime factor is 3, and its lowest power is 313^1.

    HCF(9,12,18)=3\text{HCF}(9, 12, 18) = 3
  6. Find the LCM (Least Common Multiple).

    This is the product of the highest powers of all prime factors present. - Highest power of 2 is 222^2. - Highest power of 3 is 323^2.

    LCM(9,12,18)=22×32=4×9=36\text{LCM}(9, 12, 18) = 2^2 \times 3^2 = 4 \times 9 = 36
  7. Find the product of the LCM and HCF

    Product=LCM×HCF=36×3=108\text{Product} = \text{LCM} \times \text{HCF} = 36 \times 3 = 108

    The product of the LCM and HCF is 108\boxed{108}.

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  8. Identify the given values for simple interest

    - Principal (P) = ₦30,000.00
    - Time (T) = $3\frac{1}{2}$ years = 3.5 years
    - Rate (R) = 5% per annum = 0.05
    
  9. Use the formula for simple interest

    Simple Interest (SI)=P×R×T100\text{Simple Interest (SI)} = \frac{P \times R \times T}{100}
  10. Substitute the values into the formula

    SI=30000×5×3.5100\text{SI} = \frac{30000 \times 5 \times 3.5}{100}
  11. Perform the multiplication in the numerator

    SI=30000×17.5100\text{SI} = \frac{30000 \times 17.5}{100} SI=525000100\text{SI} = \frac{525000}{100}
  12. Perform the division

    SI=₦5250.00\text{SI} = \text{₦}5250.00

    The simple interest is ₦5250.00\boxed{\text{₦}5250.00}.

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  13. Calculate the loss amount.

    Loss=Cost Price−Selling Price\text{Loss} = \text{Cost Price} - \text{Selling Price} Loss=₦45,000−₦30,000=₦15,000\text{Loss} = \text{₦}45,000 - \text{₦}30,000 = \text{₦}15,000
  14. Calculate the loss percentage using the formula

    Loss Percent=LossCost Price×100%\text{Loss Percent} = \frac{\text{Loss}}{\text{Cost Price}} \times 100\%
  15. Substitute the values and calculate

    Loss Percent=₦15,000₦45,000×100%\text{Loss Percent} = \frac{\text{₦}15,000}{\text{₦}45,000} \times 100\% Loss Percent=13×100%\text{Loss Percent} = \frac{1}{3} \times 100\% Loss Percent=33.33% (to 2 decimal places)\text{Loss Percent} = 33.33\% \text{ (to 2 decimal places)}

    The loss percent is 33.33%\boxed{33.33\%}.

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