College Preparatory Mathematics

Section 1.8 Application: Number/Geometry

Word problems can be tricky. Often it takes a bit of practice to convert the English sentence into a mathematical sentence. This is what we will focus on here with some basic number problems, geometry problems, and parts problems.

A few important phrases are described below that can give us clues for how to set up a problem.

• A number (or unknown, a value, etc) often becomes our variable

• Is (or other forms of is: was, will be, are, etc) often represents equals (\(=\)): "\(x \) is \(5 \)" becomes \(x=5 \)

• More than often represents addition and is usually built backwards, writing the second part plus the first: "Three more than a number" becomes \(x+3 \)

• Less than often represents subtraction and is usually built backwards as well, writing the second part minus the first: "Four less than a number" becomes \(x-4 \)

Using these key phrases, we can take a number problem and set up an equation to solve.

This same idea can be extended to a more involved problem as shown in the next example.

Another type of number problem involves consecutive numbers. Consecutive numbers are numbers that come one after the other, such as \(3\text{,}\) \(4\text{,}\) \(5\text{.}\) If we are looking for several consecutive numbers, it is important to first identify what they look like with variables before we set up the equation. This is shown in the following example.

Sometimes we will work consecutive even or odd integers, rather than just consecutive integers. When we had consecutive integers, we only had to add \(1\) to get to the next number, so we had \(x\text{,}\) \(x + 1\text{,}\) and \(x + 2\) for our first, second, and third number respectively. With even or odd numbers, they are spaced apart by two. So if we want three consecutive even numbers, if the first is \(x\text{,}\) the next number would be \(x + 2\text{,}\) then finally add two more to get the third, \(x + 4\text{.}\) The same is true for consecutive odd numbers, if the first is \(x\text{,}\) the next will be \(x + 2\text{,}\) and the third would be \(x+ 4\text{.}\) It is important to note that we are still adding \(2\) and \(4\) even when the numbers are odd. This is because the phrase “odd” is referring to our \(x\text{,}\) not to what is added to the numbers. Consider the next two examples.

When we started with our first, second, and third numbers for both even and odd we had \(x\text{,}\) \(x + 2\text{,}\) and \(x + 4\text{.}\) The numbers added do not change with odd or even, it is our answer for \(x\) that will be odd or even.

Another example of translating English sentences to mathematical sentences comes from geometry. A well-known property of triangles is that all three angles will always add to \(180\text{.}\) For example, the first angle may be \(50\) degrees, the second \(30\) degrees, and the third \(100\) degrees. If you add these together, \(50 + 30 + 100 = 180\text{.}\) We can use this property to find angles of triangles.

World View Note: German mathematician Bernhart Thibaut in \(1809\) tried to prove that the angles of a triangle add to \(180\) without using Euclid’s parallel postulate (a point of much debate in math history). He created a proof, but it was later shown to have an error in the proof.

Another geometry problem involves perimeter or the distance around an object. For example, consider a rectangle has a length of \(8\) and a width of \(3\text{.}\) There are two lengths and two widths in a rectangle (opposite sides) so we add \(8 + 8 + 3 + 3 = 22\text{.}\) As there are two lengths and two widths in a rectangle an alternative to find the perimeter of a rectangle is to use the formula \(P = 2L + 2W\text{.}\) So for the rectangle of length \(8\) and width \(3\) the formula would give, \(P = 2(8) + 2(3) = 16 + 6=22\text{.}\) With problems that we will consider here the formula \(P =2L+2W\) will be used.

We have seen that it is important to start by clearly labeling the variables in a short list before we begin to solve the problem. This is important in all word problems involving variables, not just consecutive numbers or geometry problems. This is shown in the following example.

Be careful on problems such as these. Many students see the phrase “double” and believe that means we only have to divide the \(444\) by \(2\) and get $\(222\) for one or both of the prices. As you can see this will not work. By clearly labeling the variables in the original list we know exactly how to set up and solve these problems.