Mastering Proportions: Problems and Examples

A piston–cylinder device initially contains ${\displaystyle 0.07m^3}$ of nitrogen gas at 130 kPa and ${\displaystyle {180}^{\circ }}$. The nitrogen is now expanded to a pressure of 80 kPa polytropically with a polytropic exponent whose value is equal to the specific heat ratio (called isentropic expansion). Determine the final temperature and the boundary work done during this process.

Ratio and proportion: ${\displaystyle 0.8:10::x:25}$

The time in hours taken by a truck to travel 144 miles is 3
Calculate time in hours taken by a truck to travel 1,200 miles:

A. What is an equation?
B. Explain the difference between a linear and a quadratic equation in terms of degree and solution.
C. Givea concise definition of the following:
1. Rate
2 Ratio
3. Proportion

What’s the difference between a ratio and a proportion?

The Otto-cycle engine in a Mercedes-Benz SLK230 has a compression ratio of 8.8.
a) What is the ideal efficiency of the engine? Use ${\displaystyle \gamma =1.40}$
b) The engine in a Dodge Viper GT2 has a slightly higher compression ratio of 9.6. How much increase in the ideal efficiency results from this increase in the compression ratio?

To find the Gear ratio of the Driving gear, 40 teeth; driven gear, 5 teeth.

Write the following fraction proportions as ratio proportions.
${\displaystyle \frac{5}{7}=\frac{15}{21}}$

Solve a ratio problem in various ways, such as make a table, draw a diagram, use a unit rate, solve an equation, and write a proportion. The aspect ratio describes the shape of a picture. It is a ratio of the width (horizontal direction) to the height (vertical direction). A photographer has a picture with an aspect ratio of 3 to 5. Can the photographer fit the picture exactly on paper that has a width of 12 inches and a height of 20 inches? Solve this problem using the Unit Rate strategy.

Let ${\displaystyle y=f\left(x\right)}$ be a function $D=(-12,13)$ and range ${\displaystyle R=(-\mathrm{\infty },\mathrm{\infty }).}$ Find the domain D and range R for the following functions and enter your answers using interval notation. Keep in mind order of operations. Be sure your intervals are in the correct order, and enter exact answers only (no approximations).
a) ${\displaystyle y=f\left(\frac{1}{2}x\right)}$
b) ${\displaystyle y=5f\left(2x\right)-2}$

For the sample of top-rated universities, the graduation rate has mean 82.7% and standard deviation 8.3%. The student/faculty ratio has mean 11.7 and standard deviation 4.3. The correlation is -0.52.
(Note: use proportion for graduation rate.)
a. Find the equation of the least squares line for predicting graduation rate from student/faculty ratio.
Let ${\displaystyle \gamma =\alpha x+\beta }$ Find A and B.
(round to five decimal places)
B=
(round to three decimal places)
b. Find the equation of the least squares line for predicting student/faculty ratio from graduation rate.
Let ${\displaystyle \gamma =Cx+D}$. Find C and D.
C=
(round to five decimal places)
D=
(round to five decimal places)

Solve the following proportion: ${\displaystyle \frac{2}{2a-1}=\frac{3}{5(a-2)}}$
A ratio of 42 to 12 is the same as a ratio of ? to 2
Solve the following proportion: ${\displaystyle \frac{x-3}{4}=\frac{x+2}{2}}$
Solve the following equation: ${\displaystyle 8-\frac{1}{2}M=15}$
Solve the following equation: ${\displaystyle 9-(x+5)=2x-(4-x)}$

Given expression is,
${\displaystyle \frac{12}{20}=\frac{18}{30}}$
We need to determine whether the ratios are equal.

Write the following fraction proportions as ratio proportions.
${\displaystyle \frac{3}{4}=\frac{75}{100}}$

Given expression is,
${\displaystyle \frac{15}{25}=\frac{3}{5}}$
We need to determine whether the ratios are equal.

Given the proportion ${\displaystyle \frac{a}{b}=\frac{8}{15}}$, what "ratio" completes the equivalent proportion ${\displaystyle \frac{a}{8}}$

Given expression is,
${\displaystyle \frac{\frac{2}{5}}{\frac{3}{10}}=\frac{15}{10\frac{1}{2}}}$
We need to determine whether the ratios are equal.

Write the following fraction proportions as ratio proportions.
${\displaystyle \frac{36}{12}=\frac{72}{24}}$

TD Canada Trust discovers that the estimated proportion of clients defaulting on a loan, if the interest rate is lower than${\displaystyle 15\mathrm{\%}}$ , is 0.34. They also discover that the estimated proportion of clients defaulting on a loan, given the interest is greater than or equal to ${\displaystyle 15\mathrm{\%}}$ , is 0.52.
a) What is the odds ratio of defaulting given the interest rate is greater than or equal to ${\displaystyle 15\mathrm{\%}}$ relative to the interest rate is lower than ${\displaystyle 15\mathrm{\%}}$? Interpret this odds ratio.
b) If we were to analyze this data using the following logistic regression model, what are the estimates of ${\displaystyle {\beta }_0}$ and ${\displaystyle {\beta }_1}$? Showyourwork.
l${\displaystyle \log (\frac{p}{1}-p)={\beta }_0+{\beta }_{1}x}$
(Where p is the probability of defaulting on a loan and x is an indicator variable that is 1 when the interest rate is greater than or equal to ${\displaystyle 15\mathrm{\%}}$ and 0 when the interest rate is less than ${\displaystyle 15\mathrm{\%}}$)

Given expression is,
${\displaystyle \frac{15}{21}=\frac{130.2}{93}}$
We need to determine whether the ratios are equal.