Hard GCSE Algebraic proportion Questions

Write the relationship using the proportional sign

"Directly proportional to the square of $v$" is written $E \propto v^{2}$.

Replace the proportional sign with a constant of proportionality

Every proportional statement becomes an equation once the constant $k$ is included. Finding $k$ is always the first job.

Substitute the pair of values you are given

Putting $v = 12$ and $E = 36$ into the equation gives one equation in one unknown.

Work out the power or root of the given value

The relevant power of $12$ is $144$, so the equation now has $k$ as its only unknown.

Rearrange to make k the subject

Divide (or multiply) to isolate $k$ — keep everything exact, never round.

Work out the constant

The constant of proportionality is $\frac{1}{4}$.

Write the complete model

This single equation now links $E$ and $v$ for every pair of values.

Check the model reproduces the given pair

Substituting $v = 12$ back gives $E = 36$, which matches the question — the model is right.

Say why that check matters

A round trip through the original data is the fastest way to catch an arithmetic slip in $k$.

Substitute the value you are asked about

This time $E$ is known, so the equation must be solved for $v$.

Rearrange to isolate the power of the unknown

Undo the multiplication by $\frac{1}{4}$ first.

Undo the power or root

Take the positive root: a length or size in a proportion question is positive, so $v = 20$.

State the final answer

So $v = 20$.

Sense check the size of the answer

$E$ grows as $v$ grows, so a larger $v$ must give a larger $E$ — check the answer moves the right way.

Reflect on the structure

Once $k$ is known the model works in both directions: forwards to find $E$, backwards to find $v$.

Turn the words into an equation

The shape of the model is fixed by the words: $y = k \sqrt{x}$.

Substitute the pair of values

Only the constant is unknown, so one pair of values is enough.

Work out the constant

The constant of proportionality is $3 \sqrt{2}$.

Write the model and check it

Substituting $x = 2$ gives $y = 6$ — the option $y = 3 \sqrt{2} \sqrt{x}$ is the correct one.

Test the first wrong option

At $x = 2$ it gives $3 \sqrt{2}$, not $6$.

Test the second wrong option

At $x = 2$ it gives $6 \sqrt{2}$, not $6$.

Test the third wrong option

At $x = 2$ it gives $3 \sqrt{2}$, not $6$.

Test the last wrong option

At $x = 2$ it gives $4$, not $6$.

Check the shape as well as the value

A correct option must have the right shape as well as fit the data point — only $y = 3 \sqrt{2} \sqrt{x}$ does both.

Say what the constant means

For every pair of values, dividing $y$ by $\sqrt{x}$ gives the same number, $3 \sqrt{2}$.

Predict a second point from the model

The model can now be used anywhere, not just at the given point.

Check that second point rules out a wrong option

The wrong option gives $6$ there, not $6 \sqrt{2}$ — the two models disagree everywhere except, at best, a single point.

Note the usual error

Reading the words as a straight-line relationship is the most common mistake in this topic.

Summarise the method

Words to equation, data to constant, constant back into the equation: the same three moves every time.

State the answer

So the correct formula is $y = 3 \sqrt{2} \sqrt{x}$.

Write the relationship as an equation

Keep the constant $k$ in — it will cancel out later, which is exactly why the answer does not depend on it.

Replace the variable with its new value

$x$ is divided by 2, so every $x$ in the model becomes $0.5x$.

Compare the new value with the old one

The constant $k$ and the $x$ cancel, leaving a pure number: $\frac{1}{8}$. $y$ is divided by 8.

Try it with numbers to be sure

Taking $x = 2$ and then $x = 1$ gives $8k$ and $1k$ — a factor of $\frac{1}{8}$, confirming the algebra.

Note the trap

The scale factor on $y$ is not the same as the scale factor on $x$ unless $y \propto x$. The power in the model changes it.

Write the general rule for this model

Multiplying $x$ by $c$ multiplies $y$ by $c^{3}$ — that is the whole of the scale-factor idea.

Check one of the wrong statements

$y$ is divided by 2. would need a factor of $\frac{1}{2}$, not $\frac{1}{8}$, so it is false.

Check another wrong statement

$y$ is divided by 6. would need a factor of $\frac{1}{6}$, not $\frac{1}{8}$, so it is false too.

Check a third wrong statement

$y$ is multiplied by 8. needs a factor of $8$ — again not $\frac{1}{8}$.

Check the last wrong statement

$y$ is divided by 4. needs a factor of $\frac{1}{4}$, so it is wrong as well.

Try a second set of numbers

A different starting value gives the same factor $\frac{1}{8}$ — the result is independent of where you start and of $k$.

Link it to the graph

The graph of this model is a curve, not a straight line, so equal steps along $x$ never give equal steps in $y$.

Summarise the method

Always form the ratio of new to old — it turns a wordy question into one short calculation.

Restate the answer

$y$ is divided by 8. This is the statement to choose.

State the conclusion

So $y$ is divided by 8.

Write the model with its constant

A percentage-change question is a scale-factor question in disguise, so keep $k$ as a letter.

Turn the percentage into a multiplier

Decreased by 20% means multiplying $x$ by $0.8 = \frac{4}{5}$.

Substitute the multiplier into the model

Replace every $x$ with $0.8x$ and simplify.

Form the ratio of new to old

The $k$ and the $x$ cancel exactly, leaving the multiplier $\frac{25}{16}$ on $y$.

Turn the multiplier back into a percentage change

A multiplier of $1.5625$ is a n increase of $56.25$%.

Work the multiplier out as an exact fraction

As an exact fraction the multiplier on $y$ is $\frac{25}{16}$. Working in fractions avoids the rounding errors that decimals bring in.

Check with a concrete pair of numbers

The ratio of those two is $\frac{25}{16}$, matching the algebra.

Note the trap

The percentage change in $y$ is not the percentage change in $x$ — the power in the model magnifies (or inverts) it.

Explain why a decrease can follow an increase

Because $y$ is inversely proportional to the square of $x$, increasing $x$ makes $y$ smaller.

Summarise the method

Three moves: percentage to multiplier, multiplier through the model, multiplier back to a percentage.

Write the general rule

Any multiplier $c$ on $x$ gives the multiplier $\frac{1}{c^{2}}$ on $y$.

Check the answer is reasonable

The multiplier is positive and greater than 1, which agrees with a percentage increase.

Reflect on why the constant never mattered

The constant $k$ cancels in the ratio, so a percentage-change question can be answered even when $k$ is unknown.

Restate the answer

The percentage increase in $y$ is $56.25$%.

State the percentage change

So $y$ increases by $56.25$%.

Write the model with its constant

The constant $k$ is unknown, but the change in $y$ is enough to find it.

Write the first value in terms of the constant

When $x = 2$, $y = 4k$.

Write the second value in terms of the constant

When $x = 6$, $y = 36k$.

Write the change as an expression

The increase in $y$ is itself a multiple of $k$, which is what makes the constant findable.

Set the change equal to the number given

The question says $y$ increases by $96$, so that expression equals $96$.

Solve for the constant

So $k = 3$ and the model is $y = 3 x^{2}$.

Check the two values

The difference is $96$, which matches the $96$ in the question — the constant is right.

Write the complete model

This is the model that answers everything else in the question.

Say why one pair of values was not given

No single pair of values is given here — only the change. That is still enough, because the change is also a multiple of $k$.

Check the model at a third value

The finished model can be used at any value, which is the point of finding $k$.

Substitute the value asked for

This time $y$ is known, so solve for $x$.

Undo the power and take the positive root

So $x = 6$ when $y = 108$.

Note the trap

Treating the relationship as a straight line would give a different constant and a wrong answer.

Sense check the direction

$y$ grows as $x$ grows, so a larger $x$ must give a larger $y$ — check the answer moves the right way.

State the final answer

So $x = 6$.