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Chapter 4: Problem 25

In Exercises \(25-60\) solve the problem by first setting up a proportion or an equation. Round off your answers to the nearest hundredth. A jar contains marbles in the ratio of 7 red to 5 black. If there are 210 black marbles, how many red marbles are there?

Short Answer

Expert verified
There are 294 red marbles.

Step by step solution

01

Set up the Ratio

The ratio of red to black marbles is given as 7 to 5. This can be written as a fraction: \(\frac{7}{5}\).
02

Define the Variables

Let the number of red marbles be \(r\) and the number of black marbles be \(b = 210\). We need to find \(r\).
03

Set Up the Proportion

Using the ratio, set up the proportion: \[\frac{r}{210} = \frac{7}{5}\].
04

Solve for \(r\)

To solve for \(r\), cross-multiply and solve the equation: \[5r = 210 \times 7\].
05

Calculate the Solution

Perform the multiplication and division: \[5r = 1470\], so \(r = \frac{1470}{5} = 294\).
06

Round the Answer

The answer is already an integer, so no rounding is needed. The number of red marbles \(r\) is 294.

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Key Concepts

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

ratios
Ratios are a way to compare two quantities. They tell us how much of one thing there is compared to another. For instance, in a jar with marbles, the ratio of 7 red marbles to 5 black marbles means that for every 7 red marbles, there are 5 black ones. A ratio can be written in three ways: as '7 to 5', '7:5', or as a fraction \(\frac{7}{5}\).
Ratios are useful in many real-life situations like cooking recipes, mixing solutions, or comparing quantities. They help us understand the relationship between different quantities and make decisions based on that relationship.
cross multiplication
When dealing with proportions, cross multiplication is a powerful tool to solve for unknowns. A proportion is an equation that states that two ratios are equal. In our problem, the proportion is set up as \(\frac{r}{210} = \frac{7}{5}\).
To solve this using cross multiplication, we multiply the numerator of one ratio by the denominator of the other ratio. This means we do: \ 5r = 210 \times 7 \. This is the 'cross' part of cross multiplication.
Once we set up this equation, it becomes easier to solve for the unknown. Cross multiplication is handy because it reduces the proportion to a single linear equation, which is much simpler to handle.
solving equations
Solving equations is a fundamental skill in math. An equation is a statement that two expressions are equal. Once we have our equation \ 5r = 210 \times 7 \, the next step is to solve for the unknown variable \(r\).
First, calculate the multiplication on the right: \ 210 \times 7 = 1470 \. Now the equation looks like this: \ 5r = 1470 \. To isolate \(r\), divide both sides by 5: \ r = \frac{1470}{5} = 294 \
This method of isolating the variable helps us find the value of \(r\). Solving equations often involves similar steps: simplifying, factoring, or using basic operations like addition, subtraction, multiplication, or division to find the solution.

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Most popular questions from this chapter

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. If David walks at the rate of \(5 \mathrm{mph}\) and jogs at the rate of \(9 \mathrm{mph}\) and it takes 2 hours to cover a distance of 16 miles, how much time was spent jogging?

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. A person drives from town A to town B at the rate of 50 mph and then flies back at the rate of 160 mph. If the total traveling time is 21 hours, how far is it from town A to town B?

If the exercise is an equation, solve it; if not, perform the indicated operations and express your answer as a single fraction. $$\frac{x+5}{2}-\frac{x-1}{4}$$

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. The length of a rectangle is 1 less than twice the width. If the width is 1 more than one seventh of the perimeter, find the dimensions of the rectangle.

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. John and Susan leave their homes at 8: 00 A.M., going toward each other along the same route, which is 9 miles long. John walks at 4 mph while Susan jogs at 8 mph. At what time will they meet?
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