Secondary 4 ratio and proportion questions become much easier when students identify what stays constant. This Mathematics tuition guide for Punggol families explains ratio, direct proportion, inverse proportion and the constant of proportionality through original worked examples.
A student may know that two quantities “go up together” but still be unsure whether the relationship is directly proportional. Another may recognise inverse proportion but forget to preserve the product. The useful question is not simply “which formula?” It is “what relationship remains unchanged?”
At eduKatePunggol, Secondary 4 Mathematics is taught in focused groups of up to three students in 1.5-hour lessons near Punggol MRT. This guide supports the wider Secondary 4 Mathematics year plan. Match the examples to the student’s actual subject level and school programme.
A ratio compares quantities in the same chosen order
If a class has 12 boys and 18 girls, the ratio boys:girls is:
12:18 = 2:3.
The order matters. Girls:boys would be 18:12 = 3:2.
Before simplifying a ratio, make sure the quantities are expressed in compatible units. A ratio of 2 m to 50 cm should not be simplified before converting one unit.
Worked example 1: divide a quantity in a ratio
Divide $420 in the ratio 3:4.
Total parts = 3 + 4 = 7.
One part = 420/7 = 60.
Therefore the two shares are:
3 × 60 = $180 and 4 × 60 = $240.
Check: $180 + $240 = $420.
Direct proportion means the ratio stays constant
If y is directly proportional to x, then:
y = kx
for some constant k.
This means y/x remains constant whenever x is non-zero.
Worked example 2: find the constant in direct proportion
y is directly proportional to x. When x = 6, y = 18. Find y when x = 11.
Since y = kx:
18 = 6k, so k = 3.
Therefore y = 3x.
When x = 11:
y = 3(11) = 33.
The important step is finding the constant first. Once k is known, the whole relationship becomes visible.
Direct proportion passes through the origin
A graph of y = kx is a straight line through the origin.
This gives a useful diagnostic. The relationship y = 3x + 2 is linear, but it is not directly proportional because when x = 0, y = 2 rather than 0.
Students sometimes confuse “straight line” with “direct proportion”. The missing condition is that the line must pass through the origin.
Inverse proportion means the product stays constant
If y is inversely proportional to x, then:
y = k/x
so xy = k.
As x increases, y decreases in a way that keeps the product constant.
Worked example 3: inverse proportion
y is inversely proportional to x. When x = 4, y = 15. Find y when x = 10.
Since xy = k:
k = 4 × 15 = 60.
Therefore y = 60/x.
When x = 10:
y = 60/10 = 6.
A useful check is that x increased from 4 to 10, so y should fall. The answer 6 fits that direction.
Worked example 4: proportional to a power
Suppose y is directly proportional to x². Then:
y = kx².
If x = 3 and y = 45:
45 = 9k, so k = 5.
When x = 4:
y = 5(16) = 80.
The word “proportional to x²” changes the relationship. Students should translate the sentence into an equation before substituting values.
Worked example 5: inverse proportion to a square
Suppose y is inversely proportional to x²:
y = k/x².
If x doubles, x² becomes four times as large, so y becomes one quarter as large.
This is why understanding the power is more reliable than memorising “inverse means divide”.
Ratio and proportion connect to scale factors
Similarity, map scales and unit conversions all use ratio structures.
If a drawing uses scale 1:500, then 1 unit on the drawing represents 500 of the same units in reality.
The similarity and scale-factors guide develops the geometric version of this reasoning.
How we diagnose proportion mistakes
Order error: a ratio is reversed halfway through.
Unit error: unlike units are compared directly.
Constant error: k is not found before a second pair is calculated.
Relationship error: direct and inverse proportion are confused.
Power error: “proportional to x²” is treated as “proportional to x”.
Why the three-student format helps
In a group of up to three students, one learner can explain the ratio order, another can find k, and another can test whether the relationship should rise or fall. The tutor can see whether the equation matches the words before the arithmetic begins.
What a 90-minute lesson could look like
An illustrative lesson could use ten minutes for ratio and units, twenty minutes for direct proportion, twenty minutes for inverse proportion, twenty minutes for powers and mixed applications, and twenty minutes for independent work, error review and continuation practice.
Repair, stabilisation and extension
Repair: use whole-number ratios and simple y = kx examples. Keep the constant visible.
Stabilisation: mix direct and inverse questions so the student must identify the relationship from the wording.
Extension: introduce powers, graphs and multi-step contexts where the student must justify the proportional model.
Try a short independent set
- Divide 360 in the ratio 2:3.
- y ∝ x and y = 28 when x = 7. Find y when x = 12.
- y ∝ 1/x and y = 9 when x = 4. Find y when x = 6.
Answers: 144 and 216; 48; and 6.
What progress should look like
- ratio order remains consistent;
- units are converted before comparison;
- the constant of proportionality is found correctly;
- direct and inverse models are distinguished;
- power relationships are translated accurately;
- answers are checked against the expected direction of change.
Punggol class details and consultation inputs
eduKatePunggol Secondary Mathematics tutorials run for 1.5 hours in groups of up to three students near Punggol MRT. Confirm current availability, fees and meeting arrangements directly.
Bring the student’s subject level, examination year and recent ratio or proportion questions. A question where the arithmetic is correct but the relationship is wrong is especially useful.
Frequently asked questions
How do I know a relationship is directly proportional?
The ratio y/x stays constant and the graph y against x is a straight line through the origin.
How do I know a relationship is inversely proportional?
The product xy stays constant for y ∝ 1/x.
Find what stays constant
Return to the Secondary 4 Mathematics year plan for the wider revision runway. For repeated percentage change, use the percentages and compound-interest guide.
Keep the ratio order clear, find the constant and let the relationship guide the calculation. Families can WhatsApp eduKatePunggol with recent work to discuss a suitable next step.
