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Chapter 6: Problem 30

Solve. Round to the nearest hundredth. $$\frac{x}{2.54}=\frac{132}{640}$$

Short Answer

Expert verified
The solution for \(x\) after rounding to the nearest hundredth is approximately 0.53.

Step by step solution

01

Cross-multiply

Cross multiply by multiplying 640 by \(x\) and 132 by 2.54 to give the equation \(640x = 132 \times 2.54\).
02

Dividing by 640

I'll isolate \(x\) by dividing both sides of the equation by 640. This gives me \(x = \frac{132 \times 2.54}{640}\).
03

Calculating \(x\)

Evaluate the expression on the right hand side using a calculator.
04

Rounding to the nearest hundredth

Round the result obtained from previous step to the nearest hundredth.

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Key Concepts

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

Rounding
Rounding numbers is a method used to simplify complex figures by limiting the number of decimal places. In this exercise, we need to round the final result of \(x = \frac{132 \times 2.54}{640}\) to the nearest hundredth. The nearest hundredth means looking at two decimal digits after the decimal point.

Here鈥檚 how to do it:
  • Identify the digit at the hundredths place (second digit after the decimal point).
  • Look at the digit immediately to the right (third digit). If it is 5 or greater, increase the hundredth place digit by one. Otherwise, leave it unchanged.
For example, if you calculate \(x\) and get 0.523, you would round it to 0.52. If you had 0.526, you would round it up to 0.53. Rounding makes numbers easier to work with and provides more manageable and interpretable results.
Equations
Equations are mathematical statements that express the equality between two expressions. In this exercise, we started with the equation \(\frac{x}{2.54}=\frac{132}{640}\). Equations can have unknowns, like \(x\) in this case, which we aim to solve.

The process to solve an equation often involves rearranging terms and using various mathematical operations to isolate the unknown variable. Key steps in solving equations include:
  • Identifying the unknown and known quantities.
  • Applying arithmetic operations like addition, subtraction, multiplication, or division.
  • Rearranging the equation to isolate the unknown variable on one side of the equation.
By manipulating the equation, we can find the value of \(x\) that makes both sides equal, thus solving the equation.
Algebraic Manipulation
Algebraic manipulation involves changing an equation or expression to get it in a desired form. The process used allows us to solve for unknowns by simplifying or rearranging terms in a systematic way. In the original problem, algebraic manipulation is used to solve the equation starting as \(\frac{x}{2.54}=\frac{132}{640}\).

Steps in algebraic manipulation in this context include:
  • Cross-multiplying, which means multiplying across the equal sign diagonally, resulting in \(640x = 132 \times 2.54\).
  • Isolating the variable by performing arithmetic operations such as division, here done by dividing both sides by 640, giving \(x = \frac{132 \times 2.54}{640}\).
  • Simplifying the expression to find a numerical result.
Algebraic manipulation is crucial because it forms the foundation of solving all kinds of equations and allowing us to express complex relationships in math clearly and concisely.

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Most popular questions from this chapter

A computer manufacturer finds an average of 3 defective hard drives in every 100 drives manufactured. How many defective drives are expected to be found in the production of 1200 hard drives?

For \(0.75 \%,\) the percent ratio is the ratio \(\frac{0.75}{100} .\) Which of the following fractions are equivalent to this ratio? (i) \(\frac{3}{4}\) (ii) \(\frac{3}{400}\) (iii) \(\frac{75}{10,000}\)

Given that \(25 \%\) of \(x\) equals \(y,\) is \(x < y\) or is \(x > y ?\)

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