Solving Ratio Problems: A Comprehensive Guide with Examples

Understanding and solving ratio problems is a fundamental skill in mathematics, often appearing in various contexts such as business, engineering, and everyday life. In this article, we will explore a detailed method to find the ratio of P:Q:R when the given ratios are P:Q 3:4 and Q:R 6:5. We will also discuss different methods to solve these problems, ensuring a well-rounded understanding.

Introduction to Ratio Problems

Before delving into the solution, it's essential to comprehend what ratios are. A ratio expresses the proportional relationship between two or more quantities. Ratios can be written in various forms, such as fractions or colon-separated values. When dealing with multiple ratios, the challenge often lies in finding a common term that connects the different ratios.

Method 1: Equalizing the Middle Term

To find the ratio P:Q:R given P:Q 3:4 and Q:R 6:5, we can adopt a method that involves equalizing the middle term, which is Q in this case.

Step 1: Express P and Q in Terms of a Common Variable

From the given ratio P:Q 3:4, we can express P and Q as:

P 3x

Q 4x

Here, x is the common multiplier.

Step 2: Express R in Terms of Q

From the given ratio Q:R 6:5, we can express Q and R as:

Q 6y

R 5y

Here, y is another common multiplier.

Step 3: Equalize Q

To equalize Q, we set the two expressions for Q equal to each other:

4x 6y

( frac{x}{y} frac{6}{4} frac{3}{2} ) which implies ( x frac{3}{2}y )

Step 4: Substitute x back into the expressions for P and Q

Substitute ( x frac{3}{2}y ) into the expression for P:

P 3x 3 left( frac{3}{2}y right) frac{9}{2}y )

Thus, P:Q:R ( frac{9}{2}y : 6y : 5y )

Step 5: Simplify the Ratio

Eliminate the common variable y and simplify the ratio:

P:Q:R 9:12:10

The greatest common divisor (GCD) of 9, 12, and 10 is 1, so the simplest form of the ratio remains 9:12:10.

Method 2: Quick Method and Inspection

An alternative, quicker method involves directly manipulating the given ratios to make Q a common term.

Step 1: Rewrite the Given Ratios

Given P:Q 3:4 and Q:R 6:5, we can express these ratios as fractions:

P:Q (frac{3}{4})

Q:R (frac{6}{5})

Step 2: Make the Denominators the Same

The least common multiple (LCM) of the denominators 4 and 6 is 12. Thus, we re-structure the ratios as follows:

P:Q (frac{3}{4} frac{9}{12})

R:Q (frac{6}{5} frac{10}{12})

Step 3: Compare P and R Relative to Q

Since Q is now 12 in both ratios, we can compare P and R as 9:10. Therefore: