Set up the proportions a/b = 9/2 and c/b = 5/3 and cross multiply.

2a = 9b and 3c = 5b.

Next, substitute the b’s in order to express a and c in terms of each other.

10a = 45b and 27c = 45b --> 10a = 27c

Lastly, reverse cross multiply to get a and c back into a proportion.

a/c = 27/10

Since there are 5 defective radios, there are 45 nondefective radios; therefore, the ratio of non-defective to defective is 45 : 5, or 9 : 1.

The ratio of green to blue is \(\displaystyle 1:3\).

Without simplifying, the ratio of green to blue is \(\displaystyle 3:9\) (order does matter).

Since 3 and 9 are both divisible by 3, this ratio can be simplified to \(\displaystyle 1:3\).

Let \(\displaystyle 10x\) be the number of store employees, \(\displaystyle 3x\) the number of store managers, and \(\displaystyle x\) the number of corporate managers.

\(\displaystyle 10x+3x+x=126\)

\(\displaystyle 14x=126\)

\(\displaystyle x=9\)

\(\displaystyle 10x=10\times9=90\), so the number of store employees is \(\displaystyle 90\).

\(\displaystyle 3x=3\times9=27\), so the number of store managers is \(\displaystyle 27\).

\(\displaystyle x=9\), so the number of corporate managers is \(\displaystyle 9\).

Therefore, the number of employees who are either store managers or corporate managers is \(\displaystyle 27+9=36\).

For this problem, we need to find the number of employees who fall into the categories described, keeping in mind that multiple portions of the pie chart must be accommodated for. Then, we can fit them into a ratio:

For the "2 to 9 years" portion of the financial industry, include

(0.2 + 0.18)(12,000,000) = 4,560,000 workers.

For the "0 to 4 years" portion of the construction industry, include

(0.15 + 0.2)(8,000,000) = 2,800,000 workers.

Now divide and simplify to find the ratio:

4,560,000/2,800,000 = 8/5.

Since the ratios are fixed, regardless of the actual values of \(\displaystyle x\), \(\displaystyle y\), or \(\displaystyle z\), we can let \(\displaystyle 2x=5y\) and \(\displaystyle 2y=3z\)

In order to convert to a form where we can relate \(\displaystyle x\) to \(\displaystyle z\), we must set the coefficient of \(\displaystyle y\) of each ratio equal such that the ratio can be transferred. This is done most easily by finding a common multiple of \(\displaystyle 5\) and \(\displaystyle 2\) (the ratio of \(\displaystyle y\) to \(\displaystyle x\) and \(\displaystyle z\), respectively) which is \(\displaystyle 10\)

Thus, we now have \(\displaystyle 4x=10y\) and \(\displaystyle 10y=15z\).

Setting the \(\displaystyle 10y\) values equal, we get \(\displaystyle 4x=15z\), or a ratio of \(\displaystyle 4:15\)

First, begin by calculating the total number of each item that there will be at the end of the process.

Cups: \(\displaystyle 30+5=35\)

Saucers: \(\displaystyle 15-3=12\)

The ratio of cups to saucers will thus be:

\(\displaystyle 35:12\)

In order to solve this problem, create an equation that summarizes the roof repair cost for each company. Begin by composing a formula for Company A, which charges 120 dollars upfront and 15 dollars per hour of labor.

\(\displaystyle 120+(15\times \textup{hours})=\textup{Cost}\)

Now, Company B charges 95 dollars upfront and 20 dollars per hour of labor. We can write the following equation:

\(\displaystyle 95+(20\times \textup{hours})=\textup{Cost}\)

The question asks us to find how many hours of labor that a repair must take in order for Company A to be cheaper than Company B. In other words, we need to compose an inequality in which the cost of Company A is less than the cost of Company B. We will substitute the variable \(\displaystyle h\) for hours and solve. 

 \(\displaystyle 120+15h< 95+20h\)

Subtract \(\displaystyle 15h\) from each side of the inequality.

\(\displaystyle 120+15h-15h< 95+20h-15h\)

\(\displaystyle 120< 95+5h\)

Subtract 95 from both sides of the inequality.

\(\displaystyle 120-95< 95-95+5h\)

\(\displaystyle 25 < 5h\)

Divide both sides of the inequality by 5. 

\(\displaystyle \frac{25}{5}< \frac{5h}{5}\)

\(\displaystyle h> 5\)

If the repair will take more than 5 hours, Company A will be cheaper.

First, let's assign variables to the names of the individuals to represent their age in 2013. 

\(\displaystyle \text{Molly}=M\)

\(\displaystyle \text{Steve}=S\)

In 2013, Molly was three times older than Steve; therefore, we can write the following expression:

\(\displaystyle M = 3S\)

We are also told that in 2016, Molly will be two times older than Steve; thus, we can write another expression: 

\(\displaystyle M + 3 = 2(S +3)\).

We can then substitute \(\displaystyle 3S\) in for \(\displaystyle M\) in the second equation to arrive at the following:

\(\displaystyle 3S + 3 = 2(S + 3)\)

\(\displaystyle 3S + 3 = 2S + 6\)

\(\displaystyle 3S = 2S + 3\)

\(\displaystyle S = 3\)

Using the given information we can generate the following two proportions:

\(\displaystyle \frac{r}{s} = \frac{4}{9}\) and \(\displaystyle \frac{s}{t} = \frac{5}{6}\)

Next, cross-multiply each proportion to come up with the following two equations: 

\(\displaystyle 9r = 4s\) and \(\displaystyle 6s = 5t\) 

Each equation shares a term with the \(\displaystyle s\) variable. We need to make this variable equal in both equations to continue. Multiply the first equation by a factor of 3 and the second by a factor of 2, so that the \(\displaystyle s\) terms are equivalent. Let's start with the first equation.

\(\displaystyle 3(9r)=3(4s)\)

\(\displaystyle 27r=12s\)

Now, we will perform a similar operation on the second equation.

\(\displaystyle 2(6s)=2(5t)\)

\(\displaystyle 12s=10t\)

Now, we can set these equations equal to one another.

\(\displaystyle 27r=12s=12s=10t\)

Drop the equivalent \(\displaystyle s\) terms.

\(\displaystyle 27r=10 t\)

 The proportion then becomes the following: 

\(\displaystyle \frac{r}{t} = \frac{10}{27}\) or \(\displaystyle \frac{t}{r} =\frac{27}{10}\)