100% stacked bar charts · 4 marks

A farm has some cows, sheep, and chickens. Last year the cows made up 60%, the sheep 25%, and the chickens 15%. There was a decrease in the number of all three animals’ population over the year. Choose the possibilities for the change in their respective shares of the population —
(i) % of cows decreased, % of sheep decreased, % of chickens decreased
(ii) % of cows increased, % of sheep increased, % of chickens increased
(iii) % of cows remained the same, % of sheep remained the same, % of chickens remained the same
(iv) % of cows decreased, % of sheep increased, % of chickens remained the same
(v) % of cows increased, % of sheep increased, % of chickens decreased.

Answer: (iii), (iv) and (v) are possible. (i) and (ii) are impossible, because the three shares must always add up to 100%: they cannot all fall or all rise.

Step-by-step solution

Given: Last year: cows 60%, sheep 25%, chickens 15%; This year: the number of each animal is lower

Idea: Shares always total 100%, so if one share goes up another must come down (or all stay the same). A fall in every number does not decide the shares; only the relative falls do.

The shares always add up to 100%. In (i) all three fall, so their sum would be less than 100%; in (ii) all rise, so the sum would exceed 100%. Both are impossible.1 mark
Take last year as 600 cows, 250 sheep, 150 chickens (total 1000). (iii) Halve everything: 300, 125, 75 (total 500) → 60%, 25%, 15%: shares unchanged. Possible.1 mark
(iv) 330 cows, 180 sheep, 90 chickens (total 600), all fewer than before → 55%, 30%, 15%: cows down, sheep up, chickens same. Possible.1 mark
(v) 520 cows, 230 sheep, 50 chickens (total 800), all fewer → 65%, 28.75%, 6.25%: cows up, sheep up, chickens down. Possible.1 mark

Possible: (iii), (iv) and (v). Impossible: (i) and (ii), since the shares must still add up to 100%.

Check: (iv): 55 + 30 + 15 = 100 ✓; (v): 65 + 28.75 + 6.25 = 100 ✓; and in each example every animal’s number fell (330 < 600, 180 < 250, 90 < 150; 520 < 600, 230 < 250, 50 < 150) ✓.

Answer to write in the exam

Shares always add to 100%

(i) all shares fall ⇒ sum < 100% ⇒ impossible; (ii) all rise ⇒ sum > 100% ⇒ impossible

Last year 600, 250, 150. (iii) 300, 125, 75 ⇒ 60%, 25%, 15%

(iv) 330, 180, 90 ⇒ 55%, 30%, 15%; (v) 520, 230, 50 ⇒ 65%, 28.75%, 6.25%

∴ (iii), (iv) and (v) are possible

Common mistakes that cost marks

Choosing (i) because ‘all numbers decreased’. A fall in numbers is not a fall in shares.
Rejecting (v) because cows’ share cannot rise when cows decreased. It can, if the other animals decreased even more.
Giving examples where one of the animals’ numbers actually went up.

How this can come in the exam

Assertion–Reason (1 mark)

Assertion (A): If the numbers of boys and girls in a school both fall, the share of boys can still rise.
Reason (R): Shares depend on the numbers relative to the total, not on the numbers alone.

Both A and R are true, and R is the correct explanation of A.
Both A and R are true, but R is not the correct explanation of A.
A is true but R is false.
A is false but R is true.

Show answer

(A) Both A and R are true, and R is the correct explanation of A.
E.g. 500 boys, 500 girls → 450 boys, 300 girls: boys’ share rises from 50% to 60%. R explains A.

Try one yourself

A class had 20 boys and 20 girls. Now it has 18 boys and 12 girls. What happened to the numbers and to the shares?