Topics › KS4 › Ratio & proportion · Foundation & Higher
Inverse proportion
-
4 workers take 6 days to build a wall. How long would 3 workers take, working at the same rate?
A
Option A:
$4.5$ days
B
Option B:
$24$ days
C
Option C:
$7$ days
D
Option D:
$8$ days
Answer and misconceptions
A
Option A:
$4.5$ days
✗
Uses direct proportionThinks fewer workers take less time. With fewer workers the job takes longer: $4 \times 6 = 24$ worker-days, then $24 \div 3$.
B
Option B:
$24$ days
✗
Finds the total work but does not share it between the workers$4 \times 6 = 24$ is the number of days one worker would take; $3$ workers take $24 \div 3$.
C
Option C:
$7$ days
✗
Thinks one fewer worker means one more dayInverse proportion is about multiplying, not adding: the total $4 \times 6$ must stay the same.
D
Option D:
$8$ days
✓ Correct
-
$y$ is inversely proportional to $x$. When $x$ is doubled, what happens to $y$?
A
Option A:
$y$ is halved
B
Option B:
$y$ is doubled
C
Option C:
$y$ decreases by $2$
D
Option D:
$y$ becomes $\frac{1}{y}$
Answer and misconceptions
A
Option A:
$y$ is halved
✓ Correct
B
Option B:
$y$ is doubled
✗
Confuses inverse proportion with direct proportionIn direct proportion $y$ doubles; in inverse proportion $xy$ stays the same, so $y$ halves.
C
Option C:
$y$ decreases by $2$
✗
Thinks inverse proportion means subtractingInverse proportion works by multiplying and dividing: if $x$ is multiplied by $2$, $y$ is divided by $2$.
D
Option D:
$y$ becomes $\frac{1}{y}$
✗
Thinks "inverse" means take the reciprocal of $y$Inverse proportion means $xy$ stays the same; if $x$ doubles, $y$ halves.
-
$y$ is inversely proportional to $x$. When $x = 4$, $y = 10$. Find $y$ when $x = 5$.
A
Option A:
$12.5$
B
Option B:
$9$
C
Option C:
$8$
D
Option D:
$0.5$
Answer and misconceptions
A
Option A:
$12.5$
✗
Uses direct proportionUses $y = 2.5x$; for inverse proportion $y = \frac{k}{x}$, so $k = 4 \times 10 = 40$.
B
Option B:
$9$
✗
Thinks $x$ up by $1$ means $y$ down by $1$Inverse proportion is about multiplying, not adding: $x \times y$ stays the same.
C
Option C:
$8$
✓ Correct
D
Option D:
$0.5$
✗
Finds $k$ by dividing instead of multiplyingWorks out $k = 10 \div 4$; in $y = \frac{k}{x}$, $k = xy = 40$.
-
Which equation shows that $y$ is inversely proportional to $x$?
A
Option A:
$y = 12x$
B
Option B:
$y = \frac{12}{x}$
C
Option C:
$y = 12 - x$
D
Option D:
$y = \frac{x}{12}$
Answer and misconceptions
A
Option A:
$y = 12x$
✗
Confuses inverse proportion with direct proportion$y = 12x$ is direct proportion: when $x$ doubles, $y$ doubles.
B
Option B:
$y = \frac{12}{x}$
✓ Correct
C
Option C:
$y = 12 - x$
✗
Thinks any decreasing relationship is inverse proportionIn $y = 12 - x$, $y$ goes down as $x$ goes up, but doubling $x$ does not halve $y$.
D
Option D:
$y = \frac{x}{12}$
✗
Thinks dividing makes it inverse proportion$y = \frac{x}{12}$ is direct proportion: doubling $x$ doubles $y$.
-
$y$ is inversely proportional to $x$. Find the missing value. $$\begin{array}{|l|c|c|}\hline x & 2 & 6\\\hline y & 15 & ?\\\hline\end{array}$$
A
Option A:
$5$
B
Option B:
$45$
C
Option C:
$11$
D
Option D:
$30$
Answer and misconceptions
A
Option A:
$5$
✓ Correct
B
Option B:
$45$
✗
Uses direct proportionMultiplies $15$ by $3$ because $x$ was multiplied by $3$; in inverse proportion $y$ is divided by $3$.
C
Option C:
$11$
✗
Thinks $x$ up by $4$ means $y$ down by $4$Inverse proportion is about multiplying, not adding: $x \times y$ stays the same.
D
Option D:
$30$
✗
Stops at the constant of proportionalityFinds $k = 2 \times 15 = 30$ but does not divide by $6$.