If ratio of ages is 3:5 and sum is 40, find ages.

Solving Age Word Problems with Ratios

Ever tried to figure out someone's age based on clues? It's like a fun math puzzle! Let's tackle a classic age problem using ratios. We’ll learn how to set up and solve equations to find the solution. Imagine you and your friend are comparing ages, or siblings trying to remember when they were a certain age ratio.

The Problem: Ages and Ratios

Here's our problem: The ratio of two people's ages is 3:5. The sum of their ages is 40. What are their ages? This kind of problem involves ratios, which are useful ways of comparing numbers.

Understanding Ratios

A ratio shows the relationship between two or more numbers. The ratio 3:5 means that for every 3 units of one age, there are 5 units of the other. Think of it like slices of a pie: if one person gets 3 slices and the other gets 5 slices, the total pie is divided into 8 parts (3+5).

Simple example: If the ratio of red to blue marbles is 2:3 and you have 2 red marbles, you must have 3 blue marbles.

Setting Up the Equations

Let's use algebra to solve this! Let's say:

• x = the younger person's age
• y = the older person's age

Now we can write two equations:

• Ratio equation: x/y = 3/5 (The ratio of their ages)
• Sum equation: x + y = 40 (The sum of their ages)

Solving the Equations (Method 1: Substitution)

We'll use substitution. First, let's solve the ratio equation for x:

x = (3/5)y

Now, substitute this value of x into the sum equation:

(3/5)y + y = 40

Solve for y:

(8/5)y = 40

y = (5/8) * 40 = 25

Now substitute y = 25 back into x = (3/5)y:

x = (3/5) * 25 = 15

Solving the Equations (Method 2: Multiplication to Eliminate Fractions)

(Optional) Let's try another method to avoid fractions. Multiply both sides of x/y=3/5 by 5y:

5x = 3y

Solve the sum equation (x+y=40) for x (x=40-y) and substitute:

5(40-y)=3y

200-5y=3y

200=8y

y=25

Solve for x in x+y=40 when y=25: x = 15

Verification and Interpretation

Let's check our answers: 15/25 simplifies to 3/5 (correct ratio!), and 15 + 25 = 40 (correct sum!).

Therefore, the ages are 15 and 25.

Conclusion: You Solved It!

We successfully used ratios and equations to solve an age problem! Practice with similar problems to build your skills. You can try changing the ratio or the sum and see what happens.

Real-World Applications: Ratio and proportion problems show up in many real-world scenarios, from cooking (following recipes) to scaling maps, and even in finance (calculating percentages).

Next steps? Search for more "age word problems" or "ratio and proportion problems" to practice.