Solving Equations with Proportions: Simplifying a/2 b/3 c/5

In this article, we will explore the solution to a proportional equation and provide a step-by-step guide on how to simplify expressions using the relationships presented. Understanding how to manipulate and interpret proportions is a fundamental skill in algebra. We'll start by defining the problem and then move on to its solution and various methods to arrive at the answer.

Problem Statement:

Given the equation a/2 b/3 c/5 k, where k is a constant, we need to find the value of (abc/c).

Step 1: Express Variables in Terms of k

First, let's express each variable in terms of k.

a/2 k

Hence, a 2k.

b/3 k

Therefore, b 3k.

c/5 k

And, c 5k.

Verification:

By substituting these values back into the original proportions, we confirm that they are consistent:

Step 2: Simplify the Expression

Next, we will simplify the expression (abc/c) using the relationships we've established.

To find the value of (abc/c), we substitute the expressions for a, b, and c in terms of k:

(abc/c) (2k * 3k * 5k) / 5k.

Since 5k in the numerator and denominator cancel out, we are left with:

2k * 3k 6k2

Finally, we can simplify it to:

(2k * 3k) / (5k) 6k2 / 5k 6k / 5 6/5 * k 6/5.

Since we know that k 1 from the problem statement, the value simplifies to:

(abc/c) 6/5 * 1 6/5 2.

Therefore, the value of abc/c is 2.

Alternatively, we can solve this more simply by observing the relationship:

If a/2 b/3 c/5 1, then:

Hence, (abc/c) (2 * 3 * 5) / 5 30 / 5 2.

Quick Solution Method

For a quick solution, we can use the given equations to express all variables in terms of a single variable, usually c, and then substitute back to find the value.

Given a/2 c/5 1, we get:

a 2

b 3

c 5

Substituting these values into the expression:

(abc/c) (2 * 3 * 5) / 5 30 / 5 2.

Thus, the value of (abc/c) is 2.

Conclusion

Solving proportional equations and simplifying expressions involves identifying the relationships between variables and applying algebraic manipulations. In this case, we used the given proportions to find a, b, and c in terms of a constant k, and then substituted these values back into the expression to find the desired value. This method can be applied to similar problems in algebra.

Key Points: Proportional equations, algebraic manipulation, ratio and proportion.