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
  • Sketch the situation (a picture is worth a thousand words)
  • 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

water tank and pipes
Tank and pipes.
Source: Sustainable sanitation

Water is being drained out of a tank through 2 pipes at the rate of 330L/min. We know that one pipe releases 50L/min more than the other. How much do the 2 pipes drain each?

Solution A: Using one variable:

Solution B: Using 2 variables and simultaneous equations:

Example 2

builders on a beam
Builders constructing roof beams.

A beam is supported by 2 pillars. The downwards force (due to gravity) is equal to the sum of the 2 upwards forces (due to the 2 pillars). Such a system is said to be in equilibrium.

The downward force is off-center in this example, so it is not acting in the center of the beam.The downwards force is four times the first upward force, and the second of the two upward forces is `6.4\ "N"` more than the first.

What are the 3 forces? (Assume we are taking the positive magnitude of the forces only.)


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

Pillar: A support column.

Equilibrium: Forces are in "equilibrium" if they cancel each other out and there is no movement in the system.

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

Solution A: Using one variable:

Solution B: Using 3 variables and simultaneous equations:

Reader's Question

A bit confused about the above 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."

vial containing a liquid drug
Vial of liquid.
Source: Helena Liu, Flickr

Example 3

A vial contains 2 g of a drug which is required for two dosages. One of the patients is a small child and the other is a large adult who needs to get 660 mg more of the drug than the child. How much should be administered to each?

(Vocabulary: A vial is a container for storing liquid).

Solution A: Using one variable:

Solution B: Using 2 variables and simultaneous equations:

Example 4

Source: RaeAllen

The manufacturer of a scooter engine recommends a gasoline-oil fuel mixture ratio of 15 to 1. In a particular garage, we can buy pure gasoline and a gasoline-oil mixture, which is 75% gasoline.

How much gasoline and how much of the gasoline-oil mix do we need to make 8.0 L of fuel for the scooter engine?

Solution A: Using one variable:

Solution B: Using 2 variables and simultaneous equations: