Question

Sum of the digits of a two-digit number is 9. When we interchange the digts,
found that the resulting new number is greater than the original number by 27. What
is the two-digit number?
One of the two digits of a two digit number is three times the other digit. If you
interchange the digits of this two-digit number and add the resulting number to the
original number, you get 88. What is the original number?​

Answer 1

Given :

• Sum of the digits of a two-digit number is 9.
• When the digits are interchanged, the resulting new number is greater than original number by 27.

To Find :

• The original number.

Solution :

Let the digit at the tens place be x.

Let the digit at the units place be y.

Original Number = (10x + y)

Case 1 :

The sum of tens digit and the units digit is 9.

Equation :

$\longrightarrow$ $\sf{x+y=9}$

$\sf{x=9-y\:\:\:(1)}$

Case 2 :

The digits when interchanged, the new number formed is greater than original number by 27.

Reversed Number = (10y +x)

Equation :

$\longrightarrow$ $\sf{10y+x=10x+y+27}$

$\longrightarrow$ $\sf{10y-y=10x-x+27}$

$\longrightarrow$ $\sf{9y=9x+27}$

$\longrightarrow$ $\sf{9y=9(9-y) +27}$

$\longrightarrow$ $\sf{9y=81-9y+27}$

$\longrightarrow$ $\sf{9y+9y=81+27}$

$\longrightarrow$ $\sf{18y=108}$

$\longrightarrow$ $\sf{y=\dfrac{108}{18}}$

$\longrightarrow$ $\sf{y=6}$

Now, substitute y = 6 in equation (1),

$\longrightarrow$ $\sf{x=9-y}$

$\longrightarrow$ $\sf{x=9-6}$

$\longrightarrow$ $\sf{x=3}$

$\large{\boxed{\bold{Ten's\:digit\:=\:x\:=\:3}}}$

$\large{\boxed{\bold{Units\:digit\:=\:y\:=\:6}}}$

$\large{\boxed{\bold{Original\:Number\:=\:10x+y=10(3)+6=30+6=36}}}$

Solution 2 :

Given :

• One of the two digits of two digit number is three times the other digit.
• When the digits are interchanged and new number formed is added to the original number the sum is 88.

To Find :

• The original two digit number.

Solution :

Let the digit at the tens place be x.

Let the digit at the units place be y.

Original Number = (10x + y)

Case 1 :

One of the digit is 3 times the other digit.

Equation :

$\longrightarrow$ $\sf{x=3y\:\:\:(1)}$

Case 2 :

When the digits are interchanged and the new number formed is added to original number the sum is 88.

Interchanged Number = (10y + x)

Equation :

$\longrightarrow$ $\sf{10y+x+10x+y=88}$

$\longrightarrow$$\sf{10y+y+10x+x=88}$

$\longrightarrow$ $\sf{11y+11x=88}$

$\longrightarrow$ $\sf{11y+11(3y)=88}$

$\longrightarrow$ $\sf{11y+33y=88}$

$\longrightarrow$ $\sf{11(y+3y)=88}$

$\longrightarrow$ $\sf{y+3y=\dfrac{88}{11}}$

$\longrightarrow$ $\sf{4y=8}$

$\longrightarrow$ $\sf{y=\dfrac{8}{4}}$

$\longrightarrow$ $\sf{y=2}$

Substitute, y = 2 in equation (1),

$\longrightarrow$ $\sf{x=3y}$

$\longrightarrow$ $\sf{x=3(2)}$

$\longrightarrow$ $\sf{x=6}$

$\large{\boxed{\bold{Ten's\:digit\:=\:x\:=\:6}}}$

$\large{\boxed{\bold{Units\:digit\:=\:y\:=\:2}}}$

$\large{\boxed{\bold{Original\:Number\:=\:10x+y=10(6)+2=60+2=62}}}$

Answer 2

Answer:

1. The original number is 36.

2. The original number is 62.

Step-by-step explanation:

1. One digit = x

Other digit = 9 - x

Original number:

One's place = 9 - x = 9 - x × 1 = 9 - x

Ten's place = x = x × 10 = 10x

After interchanging:

One's place = x = 1 × x = x

Ten's place = 9 - x = 10(9 - x) = 90 - 10x

9 - x + 10x + 27 = x + 90 - 10x

36 + 9x = 90 - 9x

36 - 90 = -9x -9x

-54 = -18x

-18x = -54

x = -54/-18 ['minus' + 'minus' = +, so both minus is cut]

x = 3

Original number:

One digit = x = 3

Other digit = 9 - x = 9 - 3 = 6

Therefore, the original number is 36.

Check ⇒

After interchanging number is 27 more than the original number

Original number = 36

After interchanging = 63

63 - 27 = 36

36 = 36

Hence, the answer is correct.

2. One digit = 3x

Other digit = x

Original number:

One's place = 3x = 3x × 1 = 3x

Ten's place = x = 10 × x = 10x

After interchanging:

One's place = x = x × 1 = x

Ten's place = 3x = 3x × 10 = 30x

If you add this both numbers you will get 88

3x + 10x + x + 30x = 88

44x = 88

x = 88/44

x = 22/11

x = 2

Original number :

One digit = 3x = 3 × 2 = 6

Other digit = x = 2

Original number = 62

Check⇒

If you add this both numbers you will get 88.

62 + 26 = 88

88 = 88

L.H.S = R.H.S

Hence, the answer is correct.

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