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Chapter 4: Problem 15

Solve for x in the following proportions. Carry division two decimal places as necessary. $$\frac{1}{250}: 2=\frac{1}{150}: x$$

Short Answer

Expert verified
x = 3.33

Step by step solution

01

Set Up the Proportion Equation

First, write down the proportion equation given in the problem. The equation states: \(\frac{1}{250}: 2 = \frac{1}{150}: x\). This can be rewritten in the form of a fraction as \(\frac{1/250}{2} = \frac{1/150}{x}\).
02

Cross-Multiply

Cross-multiplication allows us to eliminate the fractions. By cross-multiplying, we have: \((1/250) \times x = (1/150) \times 2\). Simplifying, we get \(x = \frac{2}{150} imes 250\).
03

Calculate the Right Side

First, calculate \(\frac{2}{150}\), which equals approximately 0.013333. Now, multiply this result by 250: \(0.013333 \times 250 = 3.33325\).
04

Round the Result

As the problem specifies carrying division to two decimal places, we round our result. Thus, \(x = 3.33\).

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Key Concepts

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

Cross-Multiplication
When solving proportions, cross-multiplication is a powerful tool that simplifies equations. Imagine you have a proportion where two fractions are equal, such as \(\frac{a}{b} = \frac{c}{d}\). Cross-multiplication involves multiplying across the equation: multiply "a" by "d" and "b" by "c." The result gives you an equation without fractions:
  • \(a \times d = b \times c\)
This method works because both fractions are scaled versions of the same proportion. By eliminating the fractions, you can solve for unknowns more easily, as you did in the original problem by turning it into a simple multiplication equation.
Fraction Simplification
Fraction simplification is key in making equations cleaner and simpler to solve. In mathematics, simplifying a fraction means reducing it to its simplest form.
  • Start by identifying any common factors between the numerator and the denominator.
  • Divide both numbers by the greatest common factor (GCF).
In some problems, fractions may seem complex, but breaking them down to their simplest form makes calculations more straightforward. In the exercise, simplifying helps in understanding the scale of each fraction in the proportion.
Decimal Rounding
Rounding numbers is crucial when dealing with decimals, especially in final answers. Here's how you do it:
  • Identify the specific decimal place to which you want to round, such as the hundredths place.
  • Look at the digit immediately after. If it's 5 or greater, increase the rounding place by 1.
  • If it's less than 5, keep the number in the rounding place unchanged.
Rounding ensures that your answers are manageable and easy to read, usually aligning with practical applications where precision beyond a certain point isn't necessary. In the providing exercise, after calculating to multiple decimal places, you rounded to the hundredths place to present a clean final answer.
Equation Setup
Setting up an equation is the foundation of solving mathematical problems involving proportions. When presented with a word problem or any proportion:
  • Instance one case as a fraction.
  • Set it equal to another fraction representing the second case.
It's important to align corresponding parts of the two cases across from each other in your fractions. This was done in the original problem by setting up the fractions as \(\frac{1/250}{2} = \frac{1/150}{x}\). Establishing a clear equation helps in visualizing the relationship between numbers and sets the stage for using techniques such as cross-multiplication to find unknowns.

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Most popular questions from this chapter

Set up the following word problems as proportions and solve. Include labels in the set up and on the answer. A nurse counts a client鈥檚 pulse at 19 beats in 15 seconds. If the nurse counts the client鈥檚 pulse for 1 minute, how many beats would there be in 1 minute? (1 minute = 60 seconds)

Set up the following word problems as proportions and solve. Include labels in the set up and on the answer. At an assisted living facility, the resident-to-nurse ratio is 7 to 1. If there are 84 residents, how many nurses work at the facility? ___________________________

Set up the following word problems as proportions and solve. Include labels in the set up and on the answer. If 40 vitamin tablets (tabs) contain 5,000 milligrams (mg), how many tabs are needed for a dosage of 375 mg? ___________________________

Express the following dosages as ratios. Be sure to include the units of measure and numerical value. Do not reduce the ratio. An oral solution that contains 125 mg in each 5 mL ________________________

Set up the following word problems as proportions and solve. Include labels in the set up and on the answer. A client is receiving an intravenous medication at a rate of 6.25 milligrams (mg) per minute. After 50 minutes, how many mg of medications has the client received? __________________________
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