What should be subtracted from p and added to q so that the resulting ratio becomes 1 : 5?

Question

Prepp Team · Accepted Answer

Solving the Ratio Problem: Finding the Value to Adjust Numbers

The problem asks us to find a number that, when subtracted from 'p' and added to 'q', makes the resulting ratio equal to 1:5. Let's call this unknown number 'x'.

According to the problem statement:

The number after subtracting 'x' from 'p' is $p - x$.
The number after adding 'x' to 'q' is $q + x$.
The ratio of these new numbers is given as 1:5.

We can write this relationship as an equation:

$\dfrac{p - x}{q + x} = \dfrac{1}{5}$

Step-by-Step Solution to Find x

To find the value of 'x', we need to solve this equation. We can do this by using cross-multiplication.

Multiply the numerator of the left side by the denominator of the right side, and the denominator of the left side by the numerator of the right side:

$5 \times (p - x) = 1 \times (q + x)$

Now, distribute the numbers on both sides of the equation:

$5p - 5x = q + x$

We want to isolate 'x' on one side of the equation. Let's move all terms containing 'x' to the right side and all terms not containing 'x' to the left side.

Subtract 'q' from both sides:

$5p - q - 5x = x$

Add $5x$ to both sides:

$5p - q = x + 5x$

Combine the 'x' terms on the right side:

$5p - q = 6x$

Finally, divide both sides by 6 to solve for 'x':

$x = \dfrac{5p - q}{6}$

Comparing the Solution with Options

The value we found for 'x' is $\dfrac{5p - q}{6}$. Looking at the given options (assuming P and Q are the same as p and q in the question), we find that this matches one of the options.

The options are:

$\dfrac{P + Q}{3}$
$\dfrac{P - Q}{P + Q}$
$\dfrac{5P - Q}{6}$
$\dfrac{Q - 3P}{4}$

The calculated value, $\dfrac{5p - q}{6}$, corresponds to Option 3 when using uppercase P and Q.

Conclusion

The number that should be subtracted from p and added to q so that the resulting ratio becomes 1 : 5 is $\dfrac{5p - q}{6}$.

Revision Table: Key Steps for Ratio Problems

Step	Description	Action in this Problem
1	Identify the unknown value.	Let the value be 'x'.
2	Write expressions for the new quantities.	$p-x$ and $q+x$.
3	Set up the equation based on the given ratio.	$\dfrac{p-x}{q+x} = \dfrac{1}{5}$.
4	Solve the equation for the unknown value.	Use cross-multiplication and algebraic manipulation.
5	Verify the solution (optional but recommended).	Substitute 'x' back into the original ratio expression.

Additional Information: Understanding Ratios and Algebra

A ratio is a comparison of two quantities. It can be written as $a:b$ or $\dfrac{a}{b}$. Ratios help us understand the relative size of different quantities.

Algebra is a branch of mathematics that uses variables (like 'x', 'p', 'q') to represent unknown numbers and relationships. Setting up and solving algebraic equations is a fundamental skill in many math problems, including those involving ratios.

When solving an equation like $5p - 5x = q + x$, the goal is to isolate the variable you are solving for (in this case, 'x'). This is done by performing the same operation (addition, subtraction, multiplication, division) on both sides of the equation to maintain equality.

Cross-multiplication is a useful technique for solving equations where a fraction equals another fraction ($\dfrac{a}{b} = \dfrac{c}{d}$ becomes $ad = bc$).

Ratio problems often require translating the word problem into an algebraic equation. Carefully identifying what is being compared and what the resulting ratio should be is the first step towards setting up the correct equation.