When twice of a number added to 3 is multiplied by 5 and added to the number itself, it gives 158. What is the square of that number?
169
Understanding the Word Problem and Finding the Number
The question asks us to find the square of a specific number. To do this, we first need to identify what that number is based on the description provided in the word problem. Let's break down the problem statement into smaller, manageable steps.
Let the unknown number be represented by the variable \(x\).
- "twice of a number" means \(2 \times x\), or \(2x\).
- "twice of a number added to 3" means \(2x + 3\).
- This result "is multiplied by 5" means \(5 \times (2x + 3)\).
- This product is then "added to the number itself", which means \(5(2x + 3) + x\).
- Finally, this whole expression "gives 158", so we have the equation: \(5(2x + 3) + x = 158\).
Solving the Equation to Find the Number
Now we need to solve the linear equation we formulated:
$$5(2x + 3) + x = 158$$We can solve this step-by-step:
- First, distribute the 5 into the parenthesis: $$5 \times 2x + 5 \times 3 + x = 158$$ $$10x + 15 + x = 158$$
- Combine the like terms (\(10x\) and \(x\)): $$(10x + x) + 15 = 158$$ $$11x + 15 = 158$$
- Subtract 15 from both sides of the equation to isolate the term with \(x\): $$11x + 15 - 15 = 158 - 15$$ $$11x = 143$$
- Divide both sides by 11 to solve for \(x\): $$\frac{11x}{11} = \frac{143}{11}$$ $$x = 13$$
So, the unknown number is 13.
Calculating the Square of the Number
The question asks for the square of that number. The number we found is 13.
The square of a number is the result of multiplying the number by itself.
$$ \text{Square of the number} = (\text{number})^2 $$ $$ \text{Square of 13} = 13^2 $$ $$ 13^2 = 13 \times 13 $$ $$ 13 \times 13 = 169 $$The square of the number is 169.
Verifying with the Options
Let's check our answer against the given options:
- Option 1: 225 (which is \(15^2\)) - Incorrect
- Option 2: 289 (which is \(17^2\)) - Incorrect
- Option 3: 169 (which is \(13^2\)) - Correct
- Option 4: 121 (which is \(11^2\)) - Incorrect
Our calculated square, 169, matches Option 3.
Revision Table: Steps to Solve Number Problems
| Step | Description | Application to this problem |
|---|---|---|
| 1 | Assign a variable to the unknown number. | Let the number be \(x\). |
| 2 | Translate the word problem into an algebraic equation. | \(5(2x + 3) + x = 158\) |
| 3 | Solve the equation for the variable. | \(x = 13\) |
| 4 | Perform the final calculation requested (e.g., find the square, cube, etc.). | Calculate \(x^2 = 13^2 = 169\). |
Additional Information: Linear Equations and Squares
This problem involves setting up and solving a linear equation. A linear equation is an algebraic equation in which each term has an exponent of 1, and there are no products of variables. They are called "linear" because when graphed, they form a straight line.
Solving a linear equation typically involves isolating the variable on one side of the equation using inverse operations (addition/subtraction, multiplication/division) while maintaining equality.
The square of a number \(n\) is denoted as \(n^2\), which means \(n \times n\). For example:
- The square of 5 is \(5^2 = 5 \times 5 = 25\).
- The square of -4 is \((-4)^2 = (-4) \times (-4) = 16\).
Perfect squares are the results of squaring integer numbers (like 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, etc.). Recognizing perfect squares can be helpful in solving problems.
Important Questions from Quant Based Puzzle
A number is subtracted from 4 times of it and then, the number obtained is added to its (the resultant’s) next number. If this gives the answer as 91, what was the original number?
In a class of 72 students, the number of boys is twice the number of girls. Find the number of boys.
Two years ago, T was twice as old as P. P is thrice as old as R. In five years, P will be 29. What is the present age of T?
When a number is added to its multiple of 5 and its square, the sum of these three numbers is 91. Find the number.
If the diagonal of a square is increased by 4 cm, its area increases by 56 cm 2. Find the ratio of the new area of the square to the initial area of the square.
