Sum of two numbers is 30 and their difference is 10. Find the numbers
Last Updated :
31 Mar, 2024
Sum of two numbers is 30 and their difference is 10, then those two numbers are 20 and 10.
Problem: What 2 numbers have a sum of 30 and a difference of 10?
Solution:
Let’s assume that the numbers are two integers x and y.
The sum of two variables x and y is 30. So expressing it in the form of an equation,
x + y = 30 equation (i)
The other fact that we know is that their difference is 10. Therefore,
x – y = 10 equation(ii)
Method 1: Substitution method.
Pick any one equation of the choice, say equation (i), now keep only one variable on the left-hand side of the equation and bring the other variable to the right-hand side of the equation. Implementation,
x = 30 – y —–> equation(iii)
Represent x in terms of y, Now use this derived value of x in the second equation. That is in place of x we simply have to put 30-y.
So,
30 – y – y = 10
30 – 2y = 10
2y = 20
y = 10
Once we get the value of y we can find the value of x by putting this value of y in any of the above equations.
Let’s put it in each equation,
Equation (i),
x + y = 30
x + 10 = 30
x = 20
Equation (ii),
x – y = 10
x – 10 = 10
x = 20
The same method can be used by expressing y in terms of x, Let’s pick the second equation this time,
x – y = 10
y = x – 10 —–> equation(iv)
Putting this value of y in equation (i),
x + x – 10 = 30
2x = 40
x = 20
Therefore y = 20 – 10 (using equation iv)
y = 10
Let’s consider another method for the same.
Method 2: A better approach to solve these equations would be to directly find the values by adding or subtracting the equations.
Adding equations (i) & (ii),
x + y + x – y = 30 + 10
2x = 40
x = 20
Subtracting equation (ii) from equation (i) we get,
x + y – (x – y) = 30 – 10
or, x + y – x + y = 20
2y = 20
y = 10
Note: Subtraction of equation(1) from equation(2) is also the correct approach and will eventually give the same answer.
Similar Questions
Question 1: What two numbers have a sum of 50 and a difference of 30?
Solution:
Let the numbers be x and y, Therefore
x+y=50 equation(1)
x-y=30 equation(2)
Applying the substitution method,
x-y= 30 equation(2)
x= y+30 equation(3)
Substituting the value of x in equation (1) we get,
x+y=50
y+30+y=50
2y+30=50
2y=50-30
2y=20
y=10
Putting this value of y in equation(3),
x=10+30=40
The numbers are 40 and 10
Question 2: What two numbers have a sum of 65 and a difference of 38?
Solution:
Let the numbers be x and y, Now
x+y=65 equation(1)
x-y=38 equation(2)
Applying the second method,
Adding equation (1) and equation (2) we get,
x+y+x-y=65+38
2x=103
x=51.5
Subtracting equation(2) from equation(1) we get,
x+y-(x-y)=65-38;
2y=27
y=13.5
The numbers are 51.5 and 13.5
Question 3: What two numbers have a sum of 22 and a product of 72?
Solution:
Let the two numbers be x and y. Now,
x+y=22 equation(1)
x× y= xy =72 equation(2)
Using the substitution method in equation(2) we get,
xy=72
or, x=72/y equation(3)
Putting the substituted value of x in equation(1) we get,
72/y +y=22
(72+ y× y)/y=22
72+ y× y=22y
y× y- 22y+72=0
y× y- 4y-18y+72=0
y(y- 4)-18(y- 4)=0
(y-18)(y- 4)=0
y=18 or y=4
Either value of y is acceptable.
Let’s say, the value of y=18, then in equation(3),
x=72/18= 4
Let’s say we choose the value of y=4, then in equation(3),
x=72/4=18
So, if x=4, y=18
or, if x=18, y=4
The numbers are 18 and 4.
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