# Sum of two numbers is 30 and their difference is 10. Find the numbers

The easiest way to solve any kind of arithmetic problem in words is to break it down into equations. These equations are just the mathematical expressions used for the data, which is given in words. The problem statements are known as numerical problems, and breaking them down into mathematical form helps to solve the equations, creating equations with variables and if the number of equations is lesser or equal to the number of variables, the value of the variable is easily identified. Let’s take a look at the problem statement,

### What 2 numbers have a sum of 30 and a difference of 10?

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)

When these two equations are solved, the required numbers are obtained. It is known that both the equations are true simultaneously so they are called **simultaneous linear equations**. Solving them means finding values of x and y such that both the conditions are satisfied. There are various methods to solve these equations.

**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**

**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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