The correct answer is $\displaystyle 1.71$. Cross multiplying the equation in the question will give $\displaystyle 12(5-x)=9(x+4)$. This is simplified to $\displaystyle 60-5x=9x+36$. Combining like terms gives $\displaystyle 14x=24$. Finally, isolating $\displaystyle x$ gives $\displaystyle 24/14$ or $\displaystyle 1.71$. 

$\displaystyle \frac{1}{4}\; in=1\; mile$ is the same as $\displaystyle 1\; in = 4\; miles$

So to find out the distance between the cities

$\displaystyle 12.5\; in \cdot \frac{4\; miles}{1\;in }=50\; miles$

Let $\displaystyle M,L$ be the mass of the weight and the elongation of the spring. Then for some constant of variation $\displaystyle k$, 

$\displaystyle L = kM$

We can find $\displaystyle k$ by setting $\displaystyle M = 20,L=7.2$ from the first situation:

$\displaystyle 7.2 = k \cdot 20$

$\displaystyle k = 7.2 \div 20 = 0.36$

so $\displaystyle L = 0.36 M$

In the second situation, we set $\displaystyle M = 32$ and solve for $\displaystyle L$:

$\displaystyle L = 0.36 \cdot32 =11.52$ 

which rounds to 11.5 centimeters.

First set up the proportion:

$\displaystyle \frac{3\; parts \; yellow}{1 \; part\; red}=\frac{x \; quarts \; yellow}{2 \; quarts \; red}$

x = $\displaystyle 6 \; quarts\;yellow\; paint$

Then convert this to gallons:

$\displaystyle 6 \; quarts\; \cdot \frac{1 \; gallon}{4\; quarts}= 1.50\; gallons$

To find the percentage increase, divide the number of new books by the original amount of books:

$\displaystyle 192-160=32$

She has 32 additional new books; she originally had 160.

$\displaystyle \frac{32}{160} = \frac{16}{80} = 0.20=20\%$

To find x we need to find the direct proportion. In order to do this we need to cross multiply and divide.

$\displaystyle \frac{x}{100}=\frac{1}{4}$

From here we mulitply 100 and 1 together. This gets us 100 and now we divide 100 by 4 which results in 

$\displaystyle x=25$

If the real distance between the two cities is $\displaystyle 2400$ $\displaystyle miles$, and $\displaystyle 250$ $\displaystyle miles$ = $\displaystyle 1$ $\displaystyle inch$, then we can set up the proportional equation:

$\displaystyle \frac{1 in}{250 miles}=\frac{xin}{2400 miles}$

$\displaystyle 250x=2400$ $\displaystyle in$

$\displaystyle x = 9.6$ $\displaystyle in$

We cannot solve the first equation until we know at least one of the variables, so let's solve the second equation first to solve for $\displaystyle \small y$. We therefore get:

$\displaystyle \small \frac{224}{y}=\frac{14}{3}\rightarrow14y=672\rightarrow y=48$

With our $\displaystyle \small y$, we can now find x using the first equation:

$\displaystyle \small \frac{x}{108}=\frac{20}{48}\rightarrow48x=2160\rightarrow x=45$

We therefore get the correct answer of $\displaystyle \small x=45$ and $\displaystyle \small y=48$.

Let $\displaystyle M,L$ be the mass of the weight and the elongation of the spring, respectively. Then for some constant of variation $\displaystyle k$, 

$\displaystyle L = kM$.

We can find $\displaystyle k$ by setting $\displaystyle M = 33,L=6.6$:

$\displaystyle 6.6 = k \cdot 33$

$\displaystyle k = 6.6 \div 33 = 0.2$

Therefore $\displaystyle L = 0.2 M$.

Set $\displaystyle L = 10$ and solve for $\displaystyle M$:

$\displaystyle 10= 0.2 M$

$\displaystyle M = 10 \div 0.2 = 50$ kilograms

The general formula for direct proportionality is

$\displaystyle a=kb$

where $\displaystyle k$ is the proportionality constant. To find the value of this $\displaystyle k$, we plug in $\displaystyle a=24$ and $\displaystyle b=12$

$\displaystyle 24=12k$

Solve for $\displaystyle k$ by dividing both sides by 12

$\displaystyle \frac{24}{12}=\frac{12k}{12}$

$\displaystyle 2=k$

So $\displaystyle k=2$.