# Solved Problems

# Uniform-Motion Problem: Unknown Rates

Submitted by JM on Fri, 07/17/2009 - 23:31Peter 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

Submitted by JM on Sat, 07/11/2009 - 22:46When 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

Submitted by JM on Thu, 07/09/2009 - 22:45The 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?

**Solution** **Read more »**

# Mixture Problem: Chemical Solutions

Submitted by JM on Wed, 07/08/2009 - 22:55A 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.

**Read more »**

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# Geometry Problem

Submitted by JM on Tue, 07/07/2009 - 23:48**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**