Question

# Three consecutive integers are such that when they are taken in increasing order and multiplied by 2, 3 and 4 respectfully, they add up to 74. Find these numbers.

Hint: The difference between a number and its consecutive number is 1. So if we assume that the first number is x then the second number will be x+1.And similarly the third consecutive number will be x+2.

It is given that the three integers are consecutive. And we know that a number and its consecutiveness have a difference of 1.
So, if we let the first integer be x.
Then second integer $= {\text{x + 1}}$
And third integer $= ({\text{x + 1) + 1 = x + 2}}$
The three numbers in increasing order are:
x < x+1 < x+2 , Where x is an integer.
To make the equation, it is given that the numbers in increasing order are multiplied by 2, 3 and 4 respectfully and then they added up to give 74.
On multiplying the given numbers with assumed consecutive numbers and then adding them we get:
$2{\text{x + 3(x + 1) + 4(x + 2)}}$
Now, according to question we can write:
$2{\text{x + 3(x + 1) + 4(x + 2) = 74}} \\ \Rightarrow {\text{2x + 3x + 3 + 4x + 8 = 74}} \\ \Rightarrow {\text{9x + 11 = 74}} \\ \Rightarrow {\text{9x = 74 - 11}} \\ \Rightarrow {\text{9x = 63}} \\ \Rightarrow {\text{x = }}\dfrac{{63}}{9} = 7. \\ \$
So, the first integer= 7
Putting the values of x in assumed second number, we get:
Second integer =${\text{x + 1 = 7 + 1 = 8}}$ .
Similarly, putting the value of x in assumed third number, we get:
Third integer$= {\text{x + 2 = 7 + 2 = 9}}$
So, the required three consecutive integers are 7,8 and 9.

Note: For solving this type of question, you should know that if a number is given x then to find the another consecutive number, we just add 1 to the first number so that the second number becomes x+1.similarly find the third number which (x+1)+1=x+2.After assuming the number follow the information given in question to make equation. You can also recheck your answer.