Question

Find the number which, when multiplied by 3, is 32 more than itself. The sum of two consecutive integers is 55. What is the greater integer? Find three consecutive odd numbers whose sum is 225.

Answer 1

Hint: 1. In the first question, it is given that if we multiply a number by 3, the answer should be 32 more than itself. Hence, we assume the number to be \[x\] and proceed further with the general linear arithmetic equation.
2. In the second question, it is given that the summation of two consecutive numbers is 55; here, the consecutive number means one number is greater than the other by 1. So, consider the first number to be $x$, and then the second number will be $x + 1$ and proceed further with the general linear arithmetic equation.
3. In the third question, the consecutive numbers are in an odd sequence, which means the number is not divisible by 2 completely and leaves the remainder as 1. Here the numbers are 2 more than the previous numbers. So, consider the first number to be $x$ , and then the second number will be $x + 2$ , and the third number will be $x + 4$ and proceed further with the general linear arithmetic equation.

Complete step by step answer:
 Let the number be \[x\]. According to the question, thrice of $x$ is 32 more than $x$. So, the equation is:
\[
\Rightarrow 3x = 32 + x \\
\Rightarrow 3x - x = 32 \\
\Rightarrow 2x = 32 \\
\Rightarrow x = \dfrac{{32}}{2} \\
\Rightarrow x = 16 \\
 \]
Hence, the unknown number is $x = 16$.

Let the two numbers whose addition is 55 be $x$ and $x + 1$. According to the question, the sum of the numbers is 55. So, the equation is:
\[
\Rightarrow x + x + 1 = 55 \\
\Rightarrow 2x + 1 = 55 \\
\Rightarrow 2x = 54 \\
\Rightarrow x = 27 \\
 \]
Out of $x$ and $x + 1$, the greater number is $x + 1$.
Hence,
 \[
\Rightarrow x + 1 = 27 + 1 \\
\Rightarrow = 28 \\
 \]
Hence the numbers are 27 and 28, and both are consecutive and addition will lead to 55.

Let the three consecutive odd numbers be \[x,x + 2,x + 4\]. According to the question, the sum of all the consecutive odd integers is 225. So, the equation is:
\[
\Rightarrow S = 225 \\
\Rightarrow x + x + 2 + x + 4 = 225 \\
\Rightarrow 3x = 219 \\
\Rightarrow x = \dfrac{{219}}{3} \\
\Rightarrow x = 73 \\
 \]
And the other two numbers are \[x + 2 = 75\], \[x + 4 = 77\]
Hence the numbers are 73, 75, and 77, which adds up to 225.

Note: Consecutive numbers are always one more than its previous number of the sequence, and the series depends on whether it is an odd sequence or even. If the sequence id of the odd numbers then, the numbers deferred from its preceding term by 2.