Mathematics 9709/52 — May/June 2022

Walkthrough

The summation symbol $\sum$ adds up all $n$ values of the variable. The key property used here is that summation is linear: subtracting 200 from every value can be split into summing the $x$ values and then subtracting the sum of 200 taken $n$ times. Because there are $n$ values, the constant 200 appears $n$ times, so $\sum 200 = 200n$. Substituting the two given sums, $\sum(x - 200) = 446$ and $\sum x = 6846$, gives $446 = 6846 - 200n$. Rearranging: $200n = 6846 - 446 = 6400$, so $n = \frac{6400}{200} = 32$. The number of values is therefore 32.

Key Takeaways

• Summation is linear: for a constant $c$ and $n$ terms, $\sum(x - c) = \sum x - cn$.
• A constant added or subtracted from every value contributes that constant $n$ times to the total sum.
• This idea underpins coding data: shifting all values by a constant changes the sum predictably while leaving $n$ unchanged.

Common Mistakes

• Forgetting that $\sum 200 = 200n$ rather than just 200.
• Sign errors, e.g. writing $446 = 6846 + 200n$ instead of $446 = 6846 - 200n$. The mark scheme condones one sign error but the final answer must still be correct.
• Confusing which sum is given, or substituting the wrong value.

Things to Be Careful About

• The mark scheme requires forming the equation $\sum x - \sum 200 = \sum(x - 200)$, i.e. $6846 - 200n = 446$.
• Check the arithmetic: $6846 - 446 = 6400$ and $6400 \div 200 = 32$.
• Since $n$ counts the number of values, it must be a positive whole number.