Ratio, Proportion & Rates of Change
Grade 4-6Direct Proportion
Two quantities are in direct proportion when they grow or shrink together at a fixed rate, and every direct proportion problem can be solved by finding what one single unit is worth first. This lesson covers solving proportion problems using the unitary method, finding and using the constant of proportionality k in the equation y = kx, and working backwards to find x from a given value of y.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
The unitary method
To solve most direct proportion problems, find the value of a single unit by dividing, then multiply that value by whatever quantity is needed.
8 pens cost £4.80. One pen costs £4.80 ÷ 8 = £0.60, so 5 pens cost 5 × £0.60 = £3.00.
Finding the constant of proportionality
When a problem is written using algebra, such as "y is directly proportional to x", it can be modelled by the equation y = kx, where k is a fixed number called the constant of proportionality. Find k by dividing a known y-value by its matching x-value.
y is directly proportional to x, and when x = 5, y = 20. The constant is k = 20 ÷ 5 = 4, so the equation is y = 4x.
Using the equation to find a missing value
Once the equation y = kx has been found, substitute any given value of x to find the matching value of y.
Using y = 4x, when x = 8: y = 4 × 8 = 32.
Working backwards to find x
If y is given instead of x, substitute it into the equation and rearrange to solve for x, dividing by k instead of multiplying.
Using y = 7.5x, if y = 45, then 45 = 7.5x, so x = 45 ÷ 7.5 = 6.
Worked Examples
Three exam-style questions, fully solved.
8 pens cost £4.80. Find the cost of 5 pens.
Easy- 1.Find the cost of one pen: £4.80 ÷ 8 = £0.60
- 2.Multiply by the number of pens needed: 5 × £0.60
Answer: £3.00
y is directly proportional to x. When x = 5, y = 20. Find the value of y when x = 8.
Medium- 1.Find the constant of proportionality: k = 20 ÷ 5 = 4
- 2.Write the equation: y = 4x
- 3.Substitute x = 8: y = 4 × 8
Answer: 32
The energy, E, produced by a solar panel is directly proportional to the time, t, it is exposed to sunlight. In 5 hours it produces 12.5 kWh. Find how long it takes to produce 30 kWh.
Hard- 1.Find the constant of proportionality: k = 12.5 ÷ 5 = 2.5
- 2.Write the equation: E = 2.5t
- 3.Substitute E = 30 and rearrange: 30 = 2.5t, so t = 30 ÷ 2.5
Answer: 12 hours
Avoid These
The most common mistakes students make.
Finding the value of one unit correctly, but then multiplying by the wrong number when scaling up or down.
Dividing the wrong way round when finding the constant of proportionality, for example dividing x by y instead of y by x.
Finding the constant k but forgetting to substitute it back into the equation y = kx to answer the actual question.
When working backwards to find x from a given y, multiplying by k again instead of dividing by it.
Leaving the final answer as a bare number instead of including the correct units or context, such as "£105" or "12 hours".
FAQ
Questions parents and students ask.
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