Number Problems Why do we need to learn math and how is the math that we learn useful to us? Chapter 5 will begin to answer these concerns. To resolve any real world problem with math, one must first be able to not only express that problem in a form of language but also translate the problem from that language to a mathematical expression or equation. The primary difficulty with translating from English to mathematical symbols is determining what mathematical expression to replace the given English expression with. The secondary difficulty is once one determines what mathematical symbols to use, in what order should those symbols be arranged? How to translate from English expressions to mathematical expressions. In general, we have 5 typical math symbols to consider. These are: Multiply Divide Add Subtract Equal Many different key words can be used instead of multiply, divide, add, subtract, and equal. Below is a list of key words and expressions that demonstrate how one would communicate a mathematical expression in English. It is a good idea to become familiar with these key words. Hint: Pay close attention to the subtraction key words and expressions. KEY WORDS ENGLISH MATHEMATICAL EXPRESSION EXPRESSION added to 2 added to x 2 + x more than 7 more than y y + 7 increased by 3 increased by f 3 + f greater 5 greater than s s + 5 the total of the total of x and y x + y plus 6 plus m 6 + m the sum of the sum of y and 2 y + 2 subtracted by 7 subtracted by s 7 s subtracted from 8 subtracted from x x 8 less than 4 less than x x 4 less 4 less x 4 x minus m minus 12 m 12 decreased by 15 decreased by y 15 y the difference between the difference between r - s r and s times 10 times p 10p J. R. Goodlink Page 1
multiplied by x multiplied by 2 2x the product of the product of b and c bc of one-third of y 1/3 y twice, double... twice k 2k divided by 16 divided by x 16 x t the quotient of the quotient of t and 3 3 y the ratio of the ratio of y to 5 5 equals 3 equals 3 3 = 3 is twice x is 4 2x = 4 amounts to 2 less than 5 amounts to 3 5 2 = 3 represents n represents the number of cows n = cows was Jenny was 9 last year y 1 = 9 will be Fernando will be 5 next year y + 1 = 5 the square of the cube of the square of x the cube of y 2 x 3 y This information is useful in working through word problems and English expressions. A good strategy for most word problems is to read them through 3 times. The first time the problem is read, the reader is introducing themselves to the context that the problem is presented in. It is a good idea to write down important information, usually numbers, that are noticed in this step. The second time the problem is read, it should be read more carefully and then put the important information into the form of an equation. The third time the word problem is read, check to see if the equation that was written in the second step makes sense in the context of what was read. This in-depth approach may not be necessary for every problem in section 5.1 but will be extremely useful later in this chapter. Consider the following examples. J. R. Goodlink Page 2
Example 1: 3 times a number is 5. Step 1: The key pieces of information that is noticed are 3, N (our unknown number), and 5. Step 2: After reading the problem again, we try putting our key pieces of information into an equation such as: 3 x N = 5. 3 times a number is 5 3 x N = 5 Step 3: Following the third reading of the problem while checking if the equation in step 2 makes sense, it is concluded that 3 x N = 5 is indeed the correct equation. Example 2: A number subtracted by 5 is twice that number subtracted from 2. Warning: the second step will be done incorrectly to stress the importance of the third step. Step 1: Read the problem and decide that the key information is: x, 5, 2x, and 2. Step 2: After reading a second time the equation x 5 = 2x 2 is decided as an appropriate equation for our word problem. Step 3: After reading the problem a third time it is noticed that the subtraction operations are expressed differently. The key difference is the meaning of the words from and by. After checking the list of subtraction vocabulary previously given it is noted that twice that number subtracted from 2 is actually expressed as 2 2x! The resulting equation is therefore: x 5 = 2 2x J. R. Goodlink Page 3
Example 3: Maria weighed 120 pounds before she started strength training. After training she gained 12 pounds. How much does Maria weigh after strength training? Step 1: See that the problem is about weight gain and that the key pieces of information are 120 and 12 pounds. Step 2: Notice that in the previous two example we could directly translate the verbal expression to mathematical expressions but this example lacks those typical expressions. However, it can be deduced that 120 is a good starting point and that the expression gained 12 pounds is indicative of an addition. Therefore the following expression is determined: 120 + 12 This expression is a great start but it does not answer the question of the current weight. However, we know that 120 and 12 make 132. Therefore we construct the following equation which presents an answer for the question. 120 + 12 = 132 Notice that an expression does not have an equal sign but and equation always has an equal sign. Step 3: After the third reading it is noted that the equation and the answer from step 2 seem to be reasonable given the context of the problem. It should be noted that the 3 rd step is not always necessary. It is always a good idea to check your work, however, and the 3 rd step aids in that process. J. R. Goodlink Page 4
Five Part Word Problem A five part word problem asks for five pieces of information for a word problem that has an unknown quantity. The five parts that the problem asks for are: What letter is used to represent an unknown quantity? What does the letter represent? What is the equation? What is the solution to the equation? What is the solution to the word problem? Example 4: Ha likes cookies. Today she baked 7 cookies which is 5 less than the number she baked the day before. How many cookies did she bake yesterday? Below is the format to the 5 part problem that the answers will usually be presented in by the person solving the problem. letter used: what this letter represents: equation: Solution to equation: solution to word problem: The problem revolves around cookies so let C be our unknown number. In this case, C represents the number of cookies that Ha baked yesterday. The equation is 7 = C 5. 7 = C 5 7 + 5 = C 5 + 5 12 = C Therefore, the solution to the equation is 12. Note that the solution to the equation is only a number. The solution to the word problem is Ha baked 12 cookies yesterday or some similar statement in English. letter used: C what this letter represents: Number of cookies baked yesterday equation: 7 = C 5 Solution to equation: 12 solution to word problem: Ha baked 12 cookies yesterday Notice that the five part approach to a word problem organizes the information clearly. J. R. Goodlink Page 5
Supplement Practice Problems Translate each expression into a mathematical expression #1. 5 less than h #2. 3 times 2 #3. The ratio of 3 and 5 Translate into a mathematical equation and solve. #4. One fifth a number decreased by 2 is 7. What is that number? #5. 13 decreased by the sum of 3 and x will be 8. What is x? #6. The quotient of 4 and 2 subtracted from 5 was y. Find y. #7. The sum of m and 7 will amount to 24. Find m. #8. Four increased by twelve is q. What is q? More difficult problems: #9. A dog destroyed 3 couch cushions which are 2 more than the number of cushions destroyed previously. How many couch cushions has the dog destroyed? #10. This month four students were caught cheating. This is five less than the number of students caught cheating last month. How many students were caught cheating last month? Answers: #1. h - 5 #2. 3 x 2 #3. 3/5 #4. (1/5) x n 2 = 7 n = 45 #5. 13 (3 + x) = 8 x = 2 #6. 5 (4/2) = y y = 3 #7. (m + 7) = 24 m = 17 #8. 4 + 12 = q q = 16 #9. 3 = c + 2 c =1 #10. 4 = x 5 x = 9 J. R. Goodlink Page 6