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Class 9 How Quantities Combine: Understanding Data: cheat sheet
CBSE Mathematics. When groups of different sizes are combined, the result is a weighted average. The same idea gives mixtures, mark schemes and stacked bar charts.
Key ideas
- A combined average is not the simple average of the group averages unless the groups are the same size.
- The combined average always lies between the group averages, closer to the bigger group.
- In a mixture, the price or strength of the mix is a weighted average of the parts.
- Alligation gives the ratio in which two things must be mixed to reach a target value.
- With negative marking, wrong answers subtract marks, so the score can be less than the number correct times the mark.
- A stacked bar shows amounts; a 100% stacked bar shows shares. A bigger share does not always mean a bigger amount.
Remember
| Weighted average | (w₁x₁ + w₂x₂ + …) ÷ (w₁ + w₂ + …) |
| Two groups combined | (n₁m₁ + n₂m₂) ÷ (n₁ + n₂) |
| Alligation (cheaper c, dearer d, mix m) | cheaper : dearer = (d − m) : (m − c) |
| Share in a 100% bar | part ÷ total × 100% |
Watch out
- Averaging the two averages when the groups are of different sizes.
- Forgetting to subtract the marks for wrong answers.
- Reading a 100% stacked bar as if it showed actual numbers.
Worked example
A school counts term 1 as 25%, term 2 as 25% and the final exam as 50%. Rahul scores 96, 82 and 46 (out of 100). Find the overall mark, and say how much higher or lower it is than the simple average.
Working: 25% of 96 + 25% of 82 + 50% of 46 = 67.5; simple average = 224 ÷ 3 = 74.67
Answer: 67.5; 7.17 lower than the simple average
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