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Chapter 7: Problem 38

Solve. \(170 \%\) of what number is \(221 ?\)

Short Answer

Expert verified
The number is 130.

Step by step solution

01

Translate Words into Equation

The problem asks us to find a number such that 170% of it equals 221. We can translate this to the equation: \( 1.70x = 221 \). Here, \( x \) represents the number we are looking for.
02

Solve for the Variable

We need to isolate \( x \) in the equation \( 1.70x = 221 \). This is done by dividing both sides of the equation by 1.70:\[ x = \frac{221}{1.70} \].
03

Calculate the Division

Calculate the right-hand side of the equation. Dividing 221 by 1.70 gives:\[ x = \frac{221}{1.70} \approx 130 \].
04

Verify the Solution

To ensure our calculation is correct, we multiply our solution by 1.70 and check:\( 1.70 \times 130 = 221 \).Since both sides of the original equation match with our solution, \( x = 130 \) is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation Solving
Solving equations is a fundamental skill in mathematics where we determine the value of an unknown variable that satisfies the equation. In the given problem, we have the equation \( 1.70x = 221 \). The goal is to find the value of \( x \) that makes this equation true.

There are several steps that guide us through this process:
  • Identify the variable to solve for, which in this equation is \( x \).
  • Use inverse operations to isolate the variable. Here, since \( x \) is multiplied by 1.70, we divide both sides of the equation by 1.70.
  • Perform the arithmetic operation to find \( x \). In this case, it involves division: \( x = \frac{221}{1.70} \).
Through these steps, we solve for \( x \), which ensures the equation holds true and the value of \( x \) is indeed \( 130 \). Equation solving involves understanding what the equation is asking and performing the appropriate operations to find the solution.
Translating Word Problems
Translating word problems into mathematical equations is a crucial skill that transforms complex language into solvable math sentences. Let's look at how we can do this with the problem "170% of what number is 221?".

Key points in the translation involve recognizing what each part of the sentence means in mathematical terms:
  • "170%" becomes "1.70" because percentages are converted into decimals by dividing by 100.
  • "of" typically means multiplication in mathematical terms.
  • The "what number" is what we represent as our variable, here it is \( x \).
  • "is" is translated to the equals sign "=", showing equivalence.
By identifying these elements, we are able to rewrite the sentence "170% of what number is 221" into the equation \( 1.70x = 221 \). This process makes solving the problem possible by using algebraic methods.
Division Operation
Division is a mathematical operation where a number, the dividend, is divided by another number, the divisor, to yield a quotient. In solving our exercise, division plays a pivotal role in finding the value of \( x \).

Steps to successfully apply division in equation solving:
  • Ensure that you identify the divisor and the dividend correctly. In \( x = \frac{221}{1.70} \), 221 is the dividend and 1.70 is the divisor.
  • Perform the division to calculate the quotient accurately, which will be the value of \( x \).
  • Check that the divisor is never zero, as division by zero is undefined.
After performing the division \( \frac{221}{1.70} \), we get approximately \( 130 \). It's always good practice to verify results by plugging the value back into the original equation to check if it holds true, as shown in our problem's solution.

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