Improve Your Algebra Skills with Practice Problems

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

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: ${\displaystyle 9-(x+5)=2x-(4-x)}$

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)

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

A rectangular sandbox is surronded on all sides by a strip of grass that is 5 feet wide on all sides. The length of the sandbox is 4 feet more than the width, and the total area for the sandbox and the grass is 189 square feet. Find the dimensions of the sandbox.
The length of the sandbox is ___ feet and the width is ___ feet.
(Simplify your answers. Type integers or decimals rounded to the nearest hundredth as needed.)

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 ${\displaystyle \frac{a}{8}}$

Do the equations 4x - 3y = 5 and 7y + 2x = -8 form a system of linear equations? Explain.

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.
${\displaystyle \frac{36}{12}=\frac{72}{24}}$

A woman's shoe that is size x in country A is size s(x) in country B, where ${\displaystyle s\left(x\right)=x-12}$. A woman's shoe that is size x in country ${\displaystyle p\left(x\right)=x-1.8}$. Find a function that will convert country A shoe size to country C shoe size.
The expression for the function is ___.
(Use integers or decimals for any numbers in the expression.)

A manufacturer of tools, selling rechargeable drills to a chain of home improvement stores, charges $2 more per drill than its manufacturing cost, m. The chain then sells each drill for ${\displaystyle 170\mathrm{\%}}$ of the price that it paid the manufacturer. Find a function P(m) for th price at the home improvement stores.
The function for the price at the home improvement stores is given by ${\displaystyle P\left(m\right)=?}$.

In 2000, the life expectancy of females born in certain country was 79.9 years. In 2005, it was 82.1 years. Let E(t) represent the life expectancy and t the number of years after 2000. Assume that a constant rate of change exists for the model formed.
a) Find a linear function that fits the data.
b) Use the function of part (a) to predict the life expectancy of females in 2020.
a) Let E(t) represent the life expectancy and t the number of years after 2000. Write the linear function.
${\displaystyle E\left(t\right)=?}$
b) The predicted life expectancy of females in 2020 is ?

Evaluating Exponential Functions Use a calculator to evaluate the function at the indicated values. Round your answer to three decimals.
${\displaystyle h\left(x\right)=e^x;h\left(1\right),h\left(\pi \right),h(-3),h\left(\sqrt{2}\right)}$

The number of full-time and part-time employees of an online retailer increased exponentially from 69 thousand in 2011 to 306.9 thousand in 2016.
a) Let ${\displaystyle t=0}$ correspond to 2011 and ${\displaystyle t=5}$ correspond to 2016. Then t is the number of years after 2011. Use the data points (0,69) and (5,306.9) to find the exponential growth rate and fit an exponential growth function ${\displaystyle P\left(t\right)=P_0{e}^{kt}}$ to the data, where P(t) is the number of employees, in thousands, t years after 2011.
b) Use the function found in part (a) to estimate the number of employees in 2017.
c) According to this model, when will there be one million employees?
a) P(t)=?

Exponential and Logarithmic Equations Solve the equation. Find the exact solution if possible; otherwise, use a calculator to approximate to two decimals.
${\displaystyle {\log }_8(x+5)-{\log }_8(x-2)=1}$

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 ${\displaystyle 15\mathrm{\%}}$)

Compare the decimals.
8.86_________8.65

Order the following decimals from least to greatest (separated by commas). 0.22, 0.67, 0.33, 0.6, 0.43

Given two integers. What can you claim about them, when their quotient is:
1. Positive?
2. Negative?
3. =0?