Given two numbers whose sum is 56. Six times the larger is 72 more than 5 times the smaller. Find the numbers. Please help me solve this problem in a simple explanation so I can understand it.
24 and 32
Given two numbers whose sum is 56. Six times the larger is 72 more than 5 times the smaller. Find the numbers.I need a step by step explanation please. I have a test on Monday and there will be problems like this one. So I need to know how to solve it step by step. Thanks.
To solve this problem, let's break it down into a step-by-step approach:
Step 1: Assign variables
Let's assign variables to the unknown numbers. We'll call one number "x" and the other number "y." We don't know which number is larger yet, so we'll keep it general.
Step 2: Set up equations
We are given two pieces of information: the sum of the two numbers is 56, and six times the larger number is 72 more than 5 times the smaller number.
From the first piece of information, we can write the equation:
x + y = 56
From the second piece of information, we can write another equation:
6x = 5y + 72
Step 3: Solve the equations
Now, we have a system of two equations with two variables. We can solve this system by using either substitution or elimination method.
Let's use the substitution method:
From the first equation, we can isolate x:
x = 56 - y
Now substitute this expression for x in the second equation:
6(56 - y) = 5y + 72
Simplify and solve for y:
336 - 6y = 5y + 72
336 - 72 = 5y + 6y
264 = 11y
y = 24
Step 4: Find the other number
Now substitute the value of y into the first equation to find the value of x:
x + 24 = 56
x = 56 - 24
x = 32
Step 5: Check the solution
To make sure we found the correct numbers, let's check if they satisfy the given conditions.
The sum of the numbers is 32 + 24 = 56 (as stated in the problem).
Also, six times the larger number (32 * 6 = 192) is indeed 72 more than five times the smaller number (24 * 5 + 72 = 192).
So, the two numbers are 32 and 24, respectively.