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# Solved Problems

The following posts are word problems in Mathematics with a given solution. Thus, it is called "Solved Problems". To browse by topic or category, please use the drop-down menu above.

# Uniform-Motion Problem: Unknown Rates

Peter and her daughter Liza leave home at the same time in separate cars. Peter drives to his office, a distance of 28 km, and Liza drives to school, a distance of 36 km. They arrive at their destinations at the same time. What are their average rates if Peter’s average rate is 10km/hr less than his daughter’s?

Solution:
We need to determine the average rates of Liza and Peter; therefore, we make the following representations:

$\mathbf{r:}$  the number of kilometers per hour in Liza’s average rate
$\mathbf{r - 10:}$  the number of kilometers per hour in Peter’s average rate

# Uniform-Motion Problem

When a certain object travels at a uniform rate of $\mathbf{r}$ miles per hour for a time of $\mathbf{t}$ hours, then if $\mathbf{d}$ miles is the distance traveled, the formula to derive $\mathbf{d}$ is:

$d = r \cdot t$
An algebraic problem involving the use of this formula is called uniform-motion problem. This is because in Physics, the object is assumed to move with a constant rate in every second along the way.

In applying the formula, we should always remember that the units of measurement of the rate $\mathbf{r}$, time $\mathbf{t}$, and distance $\mathbf{d}$ MUST be consistent. It means that if the distance is measured in mile and time in hour, then the rate must be expressed in miles per hour.

Problem:
One runner took 3 min 45 sec to complete a race and another runner required 4 min to run the same race. The rate of the faster runner is 0.4 m/sec more than the rate of the slower runner. Find their rates.

Solution:
Since the given data for time in the problem is seconds, we choose second as a measurement of time. Because we want to determine the rates of the runner, we have the following definition:

$\mathbf{r:}$  the number of the meter per second in the rate of the rate slower runner
$\mathbf{r + 0.4:}$  the number of the meters per second in the of the faster runner

Since each runner is in the same race, they travel equal distances.  This fact will be used to obtain an equation needed to solve the problem. Using the rate formula, we can make a table of data below. Read more »

# Investment Problem

The word problem that follows can be classified as an investment problem because it is one involving income from an investment. The income can be in the from of interest, and in that case we use the formula

$I = P \cdot R$

where $\mathbf{I}$ dollars is the annual interest earned when $\mathbf{P}$ dollars is invested at a rate $\mathbf{R}$ per year. The rate is usually given a percent; thus if the rate is 8 percent, then $\mathbf{R=0.08}$.

Investment Problem

A main invested part of USD 15,000 at 12 percent and the remainder at 8 percent. If his annual income from the two investments is USD 1,456, how much does he have invested at each rate?

# Mixture Problem: Chemical Solutions

A mixture problem can involve mixing solutions containing different percents of a substance in order to obtain a solution containing a certain percent of the substance. This is usually present in real-world problems in chemistry and other related sciences.

Another kind of mixture problem for which the method of solving is similar involves mixing commodities of different values to obtain a combination worth a specific sum of money.

# Geometry Problem

Problem: Rectangular Dimensions

If a rectangle has a length that is 3 feet less than four times its width and its perimeter is 19 feet, what are the dimensions?

Solution
We wish to determine the number of feet in each dimension of the rectangle.

$\mathbf{w:}$  the number of centimeters in the width of the rectangle
$\mathbf{4w-3:}$  the number of centimeter in the length of the rectangle

Draw a Diagram