Reciprocal Rules There are special rules for the sum and difference of reciprocals. The reciprocal of 3 is ᎏ 1 3 ᎏ and the reciprocal of x is ᎏ 1 x ᎏ . ■ If x and y are not 0, then ᎏ 1 x ᎏ + ᎏ 1 y ᎏ = ᎏ x y y ᎏ + ᎏ x x y ᎏ = ᎏ y x + y x ᎏ . ■ If x and y are not 0, then ᎏ 1 x ᎏ – ᎏ 1 y ᎏ = ᎏ x y y ᎏ – ᎏ x x y ᎏ = ᎏ y x – y x ᎏ . Translating Words into Numbers The most important skill needed for word problems is being able to translate words into mathematical operations. The following will be helpful in achieving this goal by providing common examples of English phrases and their mathematical equivalents. Phrases meaning addition: increased by; sum of; more than; exceeds by. Examples A number increased by five: x + 5. The sum of two numbers: x + y. Ten more than a number: x + 10. Phrases meaning subtraction: decreased by; difference of; less than; diminished by. Examples 10 less than a number: x – 10. The difference of two numbers: x – y. Phrases meaning multiplication: times; times the sum/difference; product; of. Examples Three times a number: 3x. Twenty percent of 50: 20% ϫ 50. Five times the sum of a number and three: 5(x + 3). Phrases meaning “equals”: is; result is. Examples 15 is 14 plus 1: 15 = 14 + 1. 10 more than 2 times a number is 15: 2x + 10 = 15. – THEA MATH REVIEW– 125 Assigning Variables in Word Problems It may be necessary to create and assign variables in a word problem. To do this, first identify any knowns and unknowns. The known may not be a specific numerical value, but the problem should indicate something about its value. Then let x represent the unknown you know the least about. Examples Max has worked for three more years than Ricky. Unknown: Ricky’s work experience = x Known: Max’s experience is three more years = x + 3 Heidi made twice as many sales as Rebecca. Unknown: number of sales Rebecca made = x Known: number of sales Heidi made is twice Rebecca’s amount = 2x There are six less than four times the number of pens than pencils. Unknown: the number of pencils = x Known: the number of pens = 4x – 6 Todd has assembled five more than three times the number of cabinets that Andrew has. Unknown: the number of cabinets Andrew has assembled = x Known: the number of cabinets Todd has assembled is five more than 3 times the number Andrew has assembled = 3x + 5 Percentage Problems To solve percentage problems, determine what information has been given in the problem and fill this informa- tion into the following template: ____ is ____% of ____ Then translate this information into a one-step equation and solve. In translating, remember that is trans- lates to “=” and of translates to “ϫ”. Use a variable to represent the unknown quantity. Examples A) Finding a percentage of a given number: In a new housing development there will be 50 houses; 40% of the houses must be completed in the first stage. How many houses are in the first stage? – THEA MATH REVIEW– 126 1. Translate. ____ is 40% of 50. x is .40 ϫ 50. 2. Solve. x = .40 ϫ 50 x = 20 20 is 40% of 50. There are 20 houses in the first stage. B) Finding a number when a percentage is given: 40% of the cars on the lot have been sold. If 24 were sold, how many total cars are there on the lot? 1. Translate. 24 is 40% of ____. 24 = .40 ϫ x. 2. Solve. ᎏ . 2 4 4 0 ᎏ = ᎏ . . 4 4 0 0 x ᎏ 60 = x 24 is 40% of 60. There were 60 total cars on the lot. C) Finding what percentage one number is of another: Matt has 75 employees. He is going to give 15 of them raises. What percentage of the employees will receive raises? 1. Translate. 15 is ____% of 75. 15 = x ϫ 75. 2. Solve. ᎏ 1 7 5 5 ᎏ = ᎏ 7 7 5 5 x ᎏ .20 = x 20% = x 15 is 20% of 75. Therefore, 20% of the employees will receive raises. – THEA MATH REVIEW– 127 Problems Involving Ratio A ratio is a comparison of two quantities measured in the same units. It is symbolized by the use of a colon—x:y. Ratios can also be expressed as fractions ( ᎏ x y ᎏ ) or using words (x to y). Ratio problems are solved using the concept of multiples. Example A bag contains 60 screws and nails. The ratio of the number of screws to nails is 7:8. How many of each kind are there in the bag? From the problem, it is known that 7 and 8 share a multiple and that the sum of their product is 60. Whenever you see the word ratio in a problem, place an “x” next to each of the numbers in the ratio, and those are your unknowns. Let 7x = the number of screws. Let 8x = the number of nails. Write and solve the following equation: 7x + 8x =60 ᎏ 1 1 5 5 x ᎏ = ᎏ 6 1 0 5 ᎏ x =4 Therefore, there are (7)(4) = 28 screws and (8)(4) = 32 nails. Check: 28 + 32 = 60 screws, ᎏ 2 3 8 2 ᎏ = ᎏ 7 8 ᎏ . Problems Involving Variation Variation is a term referring to a constant ratio in the change of a quantity. ■ Two quantities are said to vary directly if their ratios are constant. Both variables change in an equal direction. In other words, two quantities vary directly if an increase in one causes an increase in the other. This is also true if a decrease in one causes a decrease in the other. Example If it takes 300 new employees a total of 58.5 hours to train, how many hours of training will it take for 800 employees? Since each employee needs about the same amount of training, you know that they vary directly. Therefore, you can set the problem up the following way: → ᎏ 5 3 8 0 . 0 5 ᎏ = ᎏ 80 x 0 ᎏ employees ᎏᎏ hours – THEA MATH REVIEW– 128 Cross-multiply to solve: (800)(58.5)= 300x ᎏ 46 3 , 0 8 0 00 ᎏ = ᎏ 3 3 0 0 0 0 x ᎏ 156 = x Therefore, it would take 156 hours to train 800 employees. ■ Two quantities are said to vary inversely if their products are constant. The variables change in opposite directions. This means that as one quantity increases, the other decreases, or as one decreases, the other increases. Example If two people plant a field in six days, how many days will it take six people to plant the same field? (Assume each person is working at the same rate.) As the number of people planting increases, the days needed to plant decreases. Therefore, the relationship between the number of people and days varies inversely. Because the field remains constant, the two products can be set equal to each other. 2 people ϫ 6 days = 6 people ϫ x days 2 ϫ 6=6x ᎏ 1 6 2 ᎏ = ᎏ 6 6 x ᎏ 2=x Thus, it would take 6 people 2 days to plant the same field. Rate Problems In general, there are three different types of rate problems likely to be encountered in the workplace: cost per unit, movement, and work-output. Rate is defined as a comparison of two quantities with different units of measure. Rate = ᎏ x y u u n n i i t t s s ᎏ Examples ᎏ d h o o ll u a r rs ᎏ , ᎏ po co u s n t d ᎏ , ᎏ m ho il u e r s ᎏ – THEA MATH REVIEW– 129 COST PER UNIT Some problems will require the calculation of unit cost. Example If 100 square feet cost $1,000, how much does 1 square foot cost? = ᎏ $ 1 1 0 , 0 00 ft 0 2 ᎏ = $10 per square foot MOVEMENT In working with movement problems, it is important to use the following formula: (Rate)(Time) = Distance Example A courier traveling at 15 mph traveled from his base to a company in ᎏ 1 4 ᎏ of an hour less than it took when the courier traveled 12 mph. How far away was his drop off? First, write what is known and unknown. Unknown: time for courier traveling 12 mph = x. Known: time for courier traveling 15 mph = x – ᎏ 1 4 ᎏ . Then, use the formula (Rate)(Time) = Distance to find expressions for the distance traveled at each rate: 12 mph for x hours = a distance of 12x miles. 15 miles per hour for x – ᎏ 1 4 ᎏ hours = a distance of 15x – ᎏ 1 4 5 ᎏ miles. The distance traveled is the same, therefore, make the two expressions equal to each other: 12x =15x – 3.75 –15x = –15x ᎏ – – 3 3 x ᎏ = ᎏ –3 – . 3 75 ᎏ x = 1.25 Be careful, 1.25 is not the distance; it is the time. Now you must plug the time into the formula (Rate)(Time) = Distance. Either rate can be used. 12x = distance 12(1.25) = distance 15 miles = distance Total cost ᎏᎏ # of square feet – THEA MATH REVIEW– 130 . the number of screws. Let 8x = the number of nails. Write and solve the following equation: 7x + 8x =60 ᎏ 1 1 5 5 x ᎏ = ᎏ 6 1 0 5 ᎏ x =4 Therefore, there are (7)(4) = 28 screws and (8) (4) = 32. problem up the following way: → ᎏ 5 3 8 0 . 0 5 ᎏ = ᎏ 80 x 0 ᎏ employees ᎏᎏ hours – THEA MATH REVIEW– 1 28 Cross-multiply to solve: (80 0)( 58. 5)= 300x ᎏ 46 3 , 0 8 0 00 ᎏ = ᎏ 3 3 0 0 0 0 x ᎏ 156 =. takes 300 new employees a total of 58. 5 hours to train, how many hours of training will it take for 80 0 employees? Since each employee needs about the same amount of training, you know that they