SOLVING WORD PROBLEMS BASED ON TWO DIGIT NUMBER

Problem 1 :

A two digit number is four times the sum of its digits and twice the product of the digits. Find the number.

Solution :

Let xy be the required two digit number.

Given : The two digit number is four times the sum of its digits.

xy = 4(x + y)

10x + y = 4x + 4y

10x - 4x + y - 4y = 0

6x - 3y = 0

2x - y = 0

y = 2x ----(1)

Given : The two digit number is four times the sum of its digits.

xy = 2 ⋅ x ⋅ y

10x + 1y = 2xy ----(2)

Substitute y = 2x.

10x + 1(2x) = 2x(2x)

10x + 2x = 4x²

12x = 4x²

x = ¹²⁄₄

x = 3

Substitute x = 3 into (1).

y = 2(3)

y = 6

xy = 36

Therefore, the two digit number is 36.

Problem 2 :

A two digit number such that the product of its digits is 21. When 36 is subtracted from the number the digits are interchanged. Find the number.

Solution :

Let xy be the two digit number.

Given : The two digit number such that the product of its digits is 21.

x ⋅ y = 21 ----(1)

Given : When 36 is subtracted from the number the digits are interchanged.

xy - 36 = yx

10x + y - 36 = 10y + x

10x - x + y - 10 y = 36

 9x - 9y = 36

Divide both sides by 9.

x - y = 4

x = y + 4 ----(2)

Substitute x = y + 4 in to (1).

(y + 4) ⋅ y = 21

y² + 4y = 21

y² + 4y - 21 = 0

y² - 3y + 7y - 21 = 0

y(y - 3) + 7(y - 3) = 0

(y - 3)(y + 7) = 0

y - 3 = 0  or  y + 7 = 0

y = 3  or  y = 7

y represents the ones place of the two digit number and it can not be negative.

So, y = 3.

Substitute y = 3 into (2).

x = 3 + 4

x = 7

xy = 73

Therefore, the two digit number is 73.

Problem 3 :

A two digit number is such that the product of its digits is 12. When 36 is added to this number the digits are interchanged. Find the numbers.

Solution :

Let xy be the required two digit number

A two digit number such that the product of its digits is 12.

x ⋅ y = 12 -----(1)

When 36 is added to the number the digits are interchanged

xy + 36 = yx

10x + y + 36 = 10y + x

9x - 9y = -36

Divide both sides by 9.

x - y = -4

x = y - 4 ----(2)

Substitute x = y - 4 into (1).

(y - 4) ⋅ y = 12

y² - 4y = 12

y² - 4y - 12 = 0

Solve by factoring.

y² - 6y + 2y - 12 = 0

y(y - 6) + 2(y - 6) = 0

(y - 6)(y + 2) = 0

y - 6 = 0  or  y + 2 = 0

y = 6  or  y = -2

y represents the ones place of the two digit number and it can not be negative.

So, y = 6.

Substitute y = 6 into (2).

x = 6 - 4

x = 2

xy = 26

Therefore, the two digit number is 26.

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