Overview
The first step in solving real-world problems mathematically is to change the problem into an equation. Sometimes, it is a matter of deciding what operations need to be used. It is also a matter of seeing how quantities relate to one another.
Operations
Words and phrases that represent addition include such things as added to, increased from, more than, or the sum of. For example, 9 added to a number could be represented algebraically as 9 + x. Words and phrases that represent subtraction include, subtracted from, less than, decreased by, or the difference between. Statements such as multiplied by, the product of, twice or three times, or a certain percent indicate multiplication, and statements such as divided by or the quotient of can indicate division. Sometimes a statement can indicate more than one operation. For example, a statement such as 5 less than 3 times a number could be represented as 3x – 5.
Figure 1: Some words that suggest mathematical operations.
Relationships
It is also important to determine the relationship between quantities. Suppose that the length of a rectangle is 5 inches longer than its width. The width of the rectangle could be represented as w, so the length is w + 5. When one quantity is given in terms of another, it is usually easier to let one variable stand for the base amount. Any letter other than x can be used, as long as what the variable stands for is clearly defined.
Using Multiplication
Suppose that one notebook costs $3.00. Then 5 notebooks would cost 5 ∙ 3 or $15.00. If one textbook costs x, then 5 copies of the same text would cost 5x. Similarly, the cost of x items at y dollars can be represented as xy. Sometimes a statement can include more than one operation. Suppose the cost to rent a truck included a daily fee of $40.00, as well as a mileage fee of 60 cents a mile. If the truck were being rented for 2 days, the total cost of the truck would be the daily fee of 40.00 ∙ 2 for the number of days, plus the mileage fee of 0.60x.
Figure 2: Using multiplication and addition to solve a problem.
Translating Applications into Equations
The word is often means is equal to. Suppose in the truck rental example, the total cost of the truck rental is $260.00. Then $80.00 + 0.60x = 260.00. The equation could then be solved as .60x = 260 – 80, or .60x = 180, or x = 180/.60 = 300 miles.
Figure 3: An example of translating an application to an equation.
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