# Three nos are in the ratio 3:4:5 . If the sum of the largest and the smallest equals the sum of the third and 52. Find the numbers ?

Dec 3, 2016

The numbers are 39, 52 and 65

#### Explanation:

The numbers are 3n,4n and 5n
We just need to find whether 3,4,5 or 6,8,10, or 9,12,15 etc
So 3n +5n= 4n+52
Simplify
8n=4n+52
Solve
4n=52
n=13
The 3 numbers are 39:52:65

Dec 3, 2016

39,52 and 65

#### Explanation:

There should be new triangle for propionate to 3:4:5
Let take x and multiple it to 3:4:5 to make new triangle
$3 x + 5 x = 4 x + 52$
$3 x + 5 x - 4 x = 52$
or
$4 x = 52$
or
$x = \frac{52}{4}$
or
$x = 13$

Put the value of x =13 in $3 x + 5 x = 4 x + 52$
$3 \cdot 13 + 5 \cdot 13 = 4 \cdot 13 + 52$
or
$39 + 65 = 52 + 52$

or
$104 = 104$

Hence the numbers are 39,52 and 65

Dec 3, 2016

39 : 52 : 65

$\textcolor{red}{\text{There is ambiguity in this question.}}$

#### Explanation:

Consider the ratios

We have 3 parts, 4 parts and finally 5 parts. This gives a total of 12 parts

Let the first number be $a$
Let the second number be $b$
Let the third number be $c$

Let the sum of all the numbers be $s$

So we have:

$a \text{ : "b" : "c" " =" " 3" : "4" : } 5$

3 parts < 4 parts < 5 parts so $\text{ } a < b < c$ and $a + b + c = s$
the first number is $a = \frac{3}{12} s$

the second number is $b = \frac{4}{12} s$

the third number is $c = \frac{5}{12} s$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Lets break down the wording of the question:

The sum of the largest and the smallest: $\text{ } \to a + c$
equals:" "->a+c=?
the sum of:" "->a+c=?+?
the third:$\text{ } \to a + c = c +$
and 52: $\text{ } \to a + c = c + 52$

$\textcolor{red}{\text{This configuration points to } a = 52}$

$\textcolor{g r e e n}{\text{There is no point in continuing until this approach is confirmed as ok}}$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~~~~~~~~$\textcolor{m a \ge n t a}{\text{Possible error in the question}}$~~~~~~~~~~~~~
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{m a \ge n t a}{\text{The line:}}$

color(magenta)("the third: "->a+c=c+

$\textcolor{g r e e n}{\text{Should read:}}$
$\textcolor{g r e e n}{\text{the second: } \to a + c = b +}$

$\textcolor{g r e e n}{\text{or}}$
$\textcolor{g r e e n}{\text{the middle: } \to a + c = b +}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Solving for : } a + c = b + 52}$

By substitution we have:

$\frac{3}{12} s + \frac{5}{12} s = \frac{4}{12} s + 52$

$\frac{8}{12} s - \frac{4}{12} s = 52$

$\frac{1}{3} s = 52$

$\implies s = 156$

$a = \frac{1}{4} \times 156 = 39$
$b = \frac{1}{3} \times 156 = 52$
$c = \frac{5}{12} \times 156 = 65$