How would you set this problem up and work it: The difference between two numbers is 16. Three times the larger number is seven times the smaller. What are the numbers?
This one also: The differnce between two numbers is 18. The sum of twice the smaller number and three times the larger is 74. What are the numbers?
just take each fact presented and express it symbolically.
#1.
m,n are the numbers
m-n=16
3m = 7n
now solve:
3m-3n=48
7n-3n=48
4n=48
n=12
so, m=28
#2.
m-n=18
2n+3m=74
2n + 3(n+18) = 74
5n + 54 = 74
5n=20
n=4
so, m=22
Five times a whole number,x, is subtracted from 62. The result is less than 40. Find the three lowest values of x
To solve the first problem, we will set up a system of equations:
Let's call the larger number "x" and the smaller number "y".
From the given information, we have two equations:
Equation 1: "The difference between two numbers is 16."
x - y = 16
Equation 2: "Three times the larger number is seven times the smaller."
3x = 7y
Now, we have a system of equations to solve. We can use substitution or elimination method to find the values of x and y.
I will solve it using the elimination method:
Step 1: Multiply Equation 1 by 7:
7(x - y) = 7(16)
7x - 7y = 112
Step 2: Rearrange Equation 2:
3x - 7y = 0
Step 3: Add the two equations together:
(7x - 7y) + (3x - 7y) = 112 + 0
10x - 14y = 112
Step 4: Simplify the equation:
10x = 14y + 112
x = (14y + 112)/10
x = 7y/5 + 56/5
Step 5: Substitute the value of x in Equation 1:
7y/5 + 56/5 - y = 16
7y + 56 - 5y = 80
2y = 24
y = 12
Step 6: Substitute the value of y in Equation 2:
3x = 7(12)
3x = 84
x = 28
So, the larger number (x) is 28 and the smaller number (y) is 12.
Now, let's move on to the second problem:
To solve the second problem, we will again set up a system of equations:
Let's call the larger number "x" and the smaller number "y".
From the given information, we have two equations:
Equation 1: "The difference between two numbers is 18."
x - y = 18
Equation 2: "The sum of twice the smaller number and three times the larger is 74."
2y + 3x = 74
Now, we will solve this system of equations using substitution method:
Step 1: Solve Equation 1 for x:
x = y + 18
Step 2: Substitute the value of x in Equation 2:
2y + 3(y + 18) = 74
2y + 3y + 54 = 74
5y + 54 = 74
5y = 20
y = 4
Step 3: Substitute the value of y in Equation 1:
x - 4 = 18
x = 18 + 4
x = 22
So, the larger number (x) is 22 and the smaller number (y) is 4.
I hope this explanation helps you understand how to set up and solve these types of problems!