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NCERT Solutions · Class 9 Mathematics How Quantities Combine: Understanding Data

33 questions · 33 still being checked

Exercise Set 10.2 1 (part 2 of 7)

  1. Exercise 1

    Savitri's marks in Kashmiri are as follows: 35\displaystyle 35 out of 50\displaystyle 50 in internal tests, 44\displaystyle 44 out of 60\displaystyle 60 in the project, and 80\displaystyle 80 out of 100\displaystyle 100 in the final exam. The annual percentage score is calculated by combining the internals, project, and final exam in the ratio 3:4:5\displaystyle 3: 4: 5. Which of the following expression(s) gives her annual score (as a percentage) in Kashmiri?
    (i)
    35×3+44×4+80×53+4+5\displaystyle \frac{35 \times 3+44 \times 4+80 \times 5}{3+4+5}
    (ii)
    (35100)×3+(44100)×4+(80100)×53+4+5\displaystyle \frac{\left(\dfrac{35}{100}\right) \times 3+\left(\dfrac{44}{100}\right) \times 4+\left(\dfrac{80}{100}\right) \times 5}{3+4+5}
    (iii)
    (3550)×3+(4460)×4+(80100)×53+4+5×100\displaystyle \frac{\left(\dfrac{35}{50}\right) \times 3+\left(\dfrac{44}{60}\right) \times 4+\left(\dfrac{80}{100}\right) \times 5}{3+4+5} \times 100
    (iv)
    (3550×100)×3+(4460×100)×4+(80100×100)×53+4+5\displaystyle \frac{\left(\dfrac{35}{50} \times 100\right) \times 3+\left(\dfrac{44}{60} \times 100\right) \times 4+\left(\dfrac{80}{100} \times 100\right) \times 5}{3+4+5}

    Not cross-checked

    This solution has not been cross-checked against the answer printed in NCERT.

    Each mark as a percentage of its own maximum: \[\frac{35}{50}\times 100 = 70,\qquad \frac{44}{60}\times 100 = \frac{220}{3},\qquad \frac{80}{100}\times 100 = 80 \] Weighted mean with weights \(\displaystyle 3, 4, 5\): \[\frac{70\times 3+\dfrac{220}{3}\times 4+80\times 5}{3+4+5} = \frac{2710/3}{12} = \frac{1355}{18}\approx 75.28\% \](i) \[\frac{35\times 3+44\times 4+80\times 5}{12} = \frac{681}{12} = 56.75 \neq 75.28 \] Raw marks are out of $\displaystyle 50$, $\displaystyle 60$ and $\displaystyle 100$, so they are not on one scale. Not correct.(ii) \[\frac{0.35\times 3+0.44\times 4+0.80\times 5}{12} = \frac{6.81}{12} = 0.5675 \neq 75.28 \] Every mark is divided by $\displaystyle 100$, not by its own maximum. Not correct.(iii) \[\frac{\dfrac{7}{10}\times 3+\dfrac{11}{15}\times 4+\dfrac{4}{5}\times 5}{12}\times 100 = \frac{271/30}{12}\times 100 = \frac{1355}{18} \] Fractions of each maximum, weighted, then converted to a percentage. Correct.(iv) \[\frac{70\times 3+\dfrac{220}{3}\times 4+80\times 5}{12} = \frac{1355}{18} \] This is the weighted mean of the percentages itself. Correct.Answer: (iii) and (iv); the annual score is \(\displaystyle \frac{1355}{18}\approx 75.28\%\).