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# simple equation

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the sum of the digits of a two digit number is 9. the fractions formed by taking 9 less than the no.. as numerator and 9 more than the number as denominator is 3/4. what is the number

Aug 18, 2017

### 2+0 Answers

#1
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Let the number = a, then:

(a-9) / (a+9)=3/4

Solve for a:
(a - 9)/(a + 9) = 3/4

Cross multiply:
4 (a - 9) = 3 (a + 9)

Expand out terms of the left hand side:
4 a - 36 = 3 (a + 9)

Expand out terms of the right hand side:
4 a - 36 = 3 a + 27

Subtract 3 a - 36 from both sides:
Answer: | a = 63

Aug 18, 2017
#2
+1

The sum of the digits of a two digit number is 9. the fractions formed by taking 9 less than the no.. as numerator and 9 more than the number as denominator is 3/4. what is the number

Let the digits be  a and b

Note that we can write any two-digit number as   10a + b

So we have this system

a  + b  = 9

[ (10a + b) - 9]  / [ (10a + b) + 9]  = 3/4

Manipulating the first equation, we have that   b = 9 - a

Subbing this into the second equation, we have that

[ (10a + 9 - a) - 9] / [ (10a + 9 - a) + 9 ] = 3/4     simplify

[ 9a] / [ 9a + 18]  = 3/4        cross-multiply

4[9a] = 3 [9a + 18 ]   simplify

36a  = 27a + 54      subtract 27a  from both sides

9a  = 54     divide both sides by 9

6  = a

And b = 9 - a  = 9 - 6  = 3

So....our number is 63

Proof that this is correct :

[63 - 9] / [63 + 9 ] =    54 / 72   =  [3 * 18 ] / [ 4 * 18 ]  =   3 / 4   Aug 18, 2017