Solving the Ratio and Addition Problem: A Comprehensive Guide

Understanding and solving ratio and proportion problems is a fundamental skill in algebra. This guide will walk you through a step-by-step process to solve a specific problem where two numbers are in the ratio 5:8 and their total is 78. We will also explore a quick method to solve a similar problem and provide a detailed solution to an advanced version of this question.

Introduction to the Problem

Let's denote the two numbers as 5x and 8x, based on the ratio of 5:8. The total of these two numbers is given as 78. This problem can be solved by setting up and solving an equation. We will break down this solution and discuss the algebraic process to find the values of x, and subsequently, the two numbers.

Step-by-Step Solution

Given that the total of the two numbers is 78, we can set up the following equation:

$5x 8x 78$

This simplifies to:

$13x 78$

Solving for x, we obtain:

$x frac{78}{13} 6$

Now we can find the two numbers:

$text{First number} 5x 5 times 6 30$

$text{Second number} 8x 8 times 6 48$

The problem now is to find the number that should be added to the second number, which is 48, so that the ratio of the two numbers becomes 1:3.

Setting Up the Equation for the New Ratio

Let y be the number that needs to be added to the second number, 48, to make the ratio 1:3:

$frac{30}{48 y} frac{1}{3}$

Cross-multiplying to solve for y:

$3 times 30 1 times (48 y)$

$90 48 y$

Solving for y:

$y 90 - 48 42$

Therefore, the number that should be added to the second number is 42.

Quick Method for Solving the Problem

A quick way to figure out the answer is to adjust the initial ratio to match the final ratio. Since the first number remains constant, adjust the second number to align with the ratio 1:3.

Initial Ratio: 5:8

Final Ratio: 5:15 (which is the same as 1:3 when simplified)

There is an increase of 7 parts in the second number (15 - 8), which is the same as the difference between the final and initial ratios.

Given that the sum of the first two numbers (13 parts) is 78, each part represents:

$78 / 13 6$

Therefore, the required 7 parts is:

$7 times 6 42$

This confirms the answer by a different method.

Conclusion and Final Answer

Using both algebraic methods and the quick method, we have determined that the number that should be added to the second number is 42 to change the ratio from 5:8 to 1:3.

The required number is 42.

By following this guide, you can solve similar problems involving ratios and proportions effectively. This comprehensive exploration provides a clear example and multiple solution methods to aid in understanding the underlying algebraic principles.