6. Applied Verbal Problems

Mathematics is really about solving problems, not just about moving letters and numbers around.

Most real-world problems are stated using words and we need to translate them into mathematical statements.

You need to:

Read the problem carefully
Estimate the solution where possible
Assign letters to the unknown quantities
Form an equation (or equations)
Solve the equation
Check your solution against your estimate and against the original statements
Write a sentence answer (include units)

For this section, you will need to know:

We will see how to solve each problem using two methods:

one variable
simultaneous equations

You can decide which method you understand best, but if you can follow the simultaneous equations method, it is probably best for future problems.

Good luck!

Example 1:

Two pipes drain an oil tank. One pipe releases 50L/min more than the other. If they release 3300 L in 10 min together, what is the drainage rate of each?

Solution A: Using one variable:

Answer

Solution B: Using 2 variables and simultaneous equations:

Answer

Example 2:

beam
Beams during construction.

In order to produce equilibrium on a particular beam, the sum of two forces must equal a third force. If the second of the two forces is 6.4 N more than the first and the third is four times the first, what are the forces?

Vocabulary

Sometimes a question will have some words which you may not know. In this question:

Equilibrium: forces are in equilibrium if they cancel each other out.

Beam: a beam is normally made of timber, steel or concrete and holds up a roof (or similar).

Reader's Question

Dale wrote:

How can the the third force and first force be three times the first force? I just don't get it. How can the first force be three times it self when it is equal to its self? It just confused me.

My reply: Reply: Yes, Dale, these problems often seem to have funny English...

Look at the second sentence:

If the second of the two forces is 6.4 N more than the first and the third is four times the first, what are the forces?

To illustrate what this means, let me just take a guess value for the first force.

Force 1: 10 N (this is my guess value)

Force 2: 16.4 N (this is because it says the second force has to be 6.4 N more than the first)

Force 3: 40 N (this is because the third force has to be four times the first).

[Note: My guess of 10 N for the first force is not the right answer, because the first condition fails ("the sum of two forces must equal the third force") since 10 + 16.4 ≠ 40.]

It may have been easier if the question said: "The second of the two forces is 6.4 N more than the first. The third force is four times the first. What are the forces?"

Solution A: Using one variable:

Answer

Solution B: Using 3 variables and simultaneous equations:

Answer

Example 3:

A vial contains 2000 mg, which is to be used for two dosages. One patient is to be administered 660 mg more than another. How much should be administered to each?

(A vial is a container for storing liquid).

Solution A: Using one variable:

Answer

Solution B: Using 2 variables and simultaneous equations:

Answer

Example 4:

An outboard engine uses a gasoline-oil fuel mixture in the ratio of 15 to 1. How much gasoline must be mixed with a gasoline-oil mixture, which is 75% gasoline, to make 8.0 L of the mixture for the outboard engine?

In this question, we must consider the percentage of gasoline in each mixture.

Solution A: Using one variable:

Answer

Solution B: Using 3 variables and simultaneous equations:

Answer