Solving Ratio and Equation Problems

This problem involves using a given ratio and a difference between two variables to find the value of one of the variables. We are given two pieces of information:

1. The ratio of P to Q is 8 to 9: $\frac{P}{Q} = \frac{8}{9}$
2. The difference between Q and P is 12: $Q - P = 12$

Our goal is to find the numerical value of P.

Step-by-Step Solution to Find the Value of P

We have a system of two equations with two variables, P and Q. We can use substitution or elimination methods to solve for P.

Method: Substitution

Let's use the substitution method. From the first equation, $\frac{P}{Q} = \frac{8}{9}$, we can express one variable in terms of the other. Cross-multiplying gives us:

$\qquad 9P = 8Q$

From the second equation, $Q - P = 12$, we can easily express Q in terms of P:

$\qquad Q = P + 12$

Now, substitute this expression for Q into the equation derived from the ratio ($9P = 8Q$):

$\qquad 9P = 8(P + 12)$

Distribute the 8 on the right side of the equation:

$\qquad 9P = 8P + 8 \times 12$

$\qquad 9P = 8P + 96$

Now, isolate P by subtracting 8P from both sides of the equation:

$\qquad 9P - 8P = 96$

$\qquad P = 96$

So, the value of P is 96.

We can also find Q by substituting P=96 back into the equation $Q = P + 12$:

$\qquad Q = 96 + 12$

$\qquad Q = 108$

Let's check if these values satisfy the original ratio $\frac{P}{Q} = \frac{8}{9}$:

$\qquad \frac{96}{108}$

We can simplify this fraction by dividing both numerator and denominator by their greatest common divisor. Both are divisible by 12:

$\qquad \frac{96 \div 12}{108 \div 12} = \frac{8}{9}$

The ratio holds true. The difference $Q - P = 108 - 96 = 12$, which also matches the given information. Therefore, our calculated value for P is correct.

Summary of the Calculation

Revision Table: Key Concepts for Ratio and Equation Problems

Additional Information: Solving Systems of Equations

Solving systems of linear equations is a fundamental skill in algebra. There are typically two main methods for solving a system of two linear equations with two variables:

• Substitution Method: As demonstrated in the solution above, this method involves isolating one variable in one equation and substituting its expression into the other equation. This reduces the system to a single equation with one variable, which can then be solved.
• Elimination Method: This method involves multiplying one or both equations by constants so that the coefficients of one variable are opposites. Adding the modified equations together then eliminates that variable, leaving a single equation with one variable to solve.

For this specific problem, the substitution method was straightforward because the second equation ($Q - P = 12$) allowed us to easily isolate Q ($Q = P + 12$) or P ($P = Q - 12$).

Understanding how to translate word problems involving ratios and differences into algebraic equations is crucial for solving them efficiently.