Learning The Language: Word Problems

STEP ONE Read the word problem and identify the important information you will need to solve the problem.

STEP TWO Identifying what type of arithmetic you will need to do Addition Subtraction Multiplication Division

Addition Addition story problems often use words like: Increased by More than Combined Together Total of Sum Added to EXAMPLE: Jane has 10 Barbie's and for her birthday she gets 3 more. How many Barbie’s does Jane have now? (10+3=?)

Subtraction Subtraction story problems often use words like: Decreased by Minus Difference Less than Fewer than Away/loose “Subtract from” EXAMPLE: If there are 10 cars in one parking and 6 less cars in the second parking lot. How many more cars are there in the second parking lot? (10-6=?)

Multiplication Multiplication story problems often use words like: Of Times Multiplied by Product of EXAMPLE: If Mary has 3 pets and Annie has 2 times as many pets as Mary. How many pets does Annie have? (3x2=?)

Division Division word problems often use words like: Per Out of Ratio of Quotient of “a” EXAMPLE: John ate a total allowance of $250. If he spends $25 a day, how many days will the allowance last? EXAMPLE: If Bobbi had 15 cookies and ate the same amount each day for 5 days how many did she eat per day? (15 / 5=? )

STEP THREE Solve the Problem Using one of the many problem solving strategies

Choose a Strategy to Solve the Problem: Working Backwards Drawings and illustrations Making an equation Visualizations Make a Table Guess and Check Or use your own strategy

WORKING BACKWARDS A problem you would use the working backward method on would be something like this: Mary Ann flew from Marquette, Mi to Los Angeles , CA . It took her 2 hours to get from Marquette to Chicago, Il and 4 hours to get from Chicago to Los Angeles. If she arrived at 4:00 pm what time was it when she left? Figure out what you are trying to find. In this case it is the time in which she left Marquette. Make a plan of action. In this case you would take the time she arrived and work backwards by subtracting the hours she was in flight. 4:00 (when she arrived in LA) – 4 hours (it took to go from Chicago to LA) = 12:00 (time she left Chicago). You would then take that time and subtract the time it took to go from Chicago to Marquette. 12:00pm – 2 hours = 10:00 am (your answer)

DRAWINGS AND ILLUSTRATIONS Drawing a picture is a great way to solve word problems. You not only get the answer but it is easy to see WHY you get the answer. A good example of a problem you would want to make a drawing for would be a problem like: For Stacie's birthday she got a bag of marbles from her friend Amy. The bag has 6 red marbles, 10 blue marbles, 4 yellow marbles, and 1 green marble. How many marbles does she have in her bag? Figure out what you are trying to find: How many marbles there are in the bag. Make a plan: Draw out each set of marbles and count them up. There are a total of 21 marbles!

MAKE AN EQUATION Making an equation of story problems is also a great way to solve story problems. You just take the numbers from the problem and turn them into an equation. This problem would be a good example of when to use an equation: For a school bake sale 5 students each brought in something to sell. Keri brought 2 dozen cookies, Rachel brought 3 dozen brownies, Max brought 5 dozen muffins, Michelle brought 1 dozen cupcakes, and Sarah brought 4 dozen rice crispy bars. How many treats did they have to sell? Decide what you are trying to find in this case: How many treats they will have to sell. Make a plan or in this case an equation. We know that there are 12 treats in a dozen and we know how many dozen cookies we have so here are some sample equations you could use: 2(12)+3(12)+5(12)+1(12)+4(12)=180 (2+3+5+1+4)12=180 Then just simply solve the Problem Mathematically

VISUALIZATIONS/HANDS ON This problem solving strategy can be the most fun and it is very simple. You actually use visuals to do the problem much like when using drawings but instead of using pencil and paper you use the actual things. Say you have a problem like this: At the beginning and the end of every day Mrs. Smith collects and hands back papers. On Monday at the beginning of the day she hands back 25 and collects 18. At the end of the day she hands back 17 and collects 15. How many papers will the teacher have collected on Monday and how many will the students have gotten back? To do this problem hands on is very simple. I would actually take the class and do exactly what the story problem says. Hand out some papers, collect some paper, and repeat the process. As if it were the beginning and end of the day. Then when you are finished count the papers the students have and how many the teacher has.

MAKE A TABLE Making a table is a very organized and simple way to solve some story problems. It is best used when dealing with problems like: Andy and his parents decided that for his allowance would go up one dollar and 50 cents every week for 3 consecutive weeks. If he starts out at getting 6 dollars how much would he make week 5? Find: What will his allowance be week 5? Plan: Make a chart of what his allowance will be each week  Week $ allowance 1 $6.00 2 $7.50 3 $9.00 4 $10.50 5 $12.00 $12.00

GUESS AND CHECK They guess and check method isn’t the fastest but it is very effective. You would usually use it on problems like this: If two sisters ages add up to 22 years and one is 4 years older than the other what are there two ages? You are trying to find what: Their Ages Plan: Select random numbers that add up to 22 until you find two that are 4 apart. 10 and 12: 10+12=22 but 12-10=2 not 4; 8 and 15: 8+15= 22 but 15- 8=6; 9 and 13: 9+13=22 and 13-9=4 so there ages are 9 and 13!

STEP FOUR Writing your answer to the story problem is the final step When writing the answer there are a few things you have to remember What are you trying to find If your answer should be in units such as (mph, cups, or inches) Your answer should be in complete sentences

Examples of Answers Wrong way to Answer this Story Problem: If Keri has 3 apples and 5 oranges how many more oranges does she have than apples? Wrong way to Answer this Story Problem: 2 (it is the right answer but when working with story problems you have to explain your answer) Right Way to Answer this Story Problem: Keri has 2 more oranges than apples. Now that you are familiar with Solving Story Problems lets test your memory with some worksheets and a quiz!

PROBLEM Read this problem and use the information to answer the questions. Dwayne Johnson’s net earnings for last month was $726. During that month he spent 10% on tithes, $150 on gas, 25% on rent, $90 on cellphone, 15% on groceries, 5% on entertainment, and $300 on his car payment.

What was Dwayne’s gross earnings last month? QUESTION #1 What was Dwayne’s gross earnings last month? $3,249 $2,813 $1,756 A B C

How much money did he spend on groceries and rent? QUESTION #2 How much money did he spend on groceries and rent? $1,125.20 $3,500.46 $937.15 A B C

QUESTION #3 If Dwayne did not pay his car payment, how much would he have in net income? $1,026 $1,214.33 $426 A B B C

Principal and Interest A total of $20,000 was invested between two accounts one paying 4% simple interest and the other paying 3% simple interest. After 1 year the total interest was $720. How much was invested at each rate? I = Prt

Using a Table Accounts P r t = I 4% x .04 1 .04x 3% 20,000 - x .03

I1 + I2 = $720 .04x + .03(20,000 – x) = 720 .04x + 600 - .03x = 720 .01x = 120 x = 12,000 Furthermore 20,000 – 12,000 = 8,000 Thus the amount invested at each rate is $12,000 at 4% and $8,000 at 3%

Mixture Problems How many ounces of 30% alcohol solution that must be mixed with 10 ounces of a 70% solution to obtain a solution that is 40% alcohol?

30% alcohol 70%alcohol 40% alcohol + =

Using a Table Alcohol Concentration Ounces Percent Solution 30% of Alcohol x .30 .30x 70% of Alcohol 10 .70 .70(10) 40% of Alcohol x + 10 .40 .40(x + 10)

.30x + .70(10) = .40(x + 10) .30x + 7 = .40x + 4 3 = .10x 30 = x Furthermore 30 +10 = 40 Thus, the amount of alcohol at each concentration is 30 ounces at 30% 40 ounces at 40%

Solving Mixture Problems Example: The owner of a candy store is mixing candy worth $6 per pound with candy worth $8 per pound. She wants to obtain 144 pounds of candy worth $7.50 per pound. How much of each type of candy should she use in the mixture? 1.) UNDERSTAND Let n = the number of pounds of candy costing $6 per pound. Since the total needs to be 144 pounds, we can use 144  n for the candy costing $8 per pound. Continued 29

Solving Mixture Problems Example continued 2.) TRANSLATE Use a table to summarize the information. Number of Pounds Price per Pound Value of Candy $6 candy n 6 6n $8 candy 144  n 8 8(144  n) $7.50 candy 144 7.50 144(7.50) 6n + 8(144  n) = 144(7.5) # of pounds of $6 candy # of pounds of $8 candy # of pounds of $7.50 candy Continued

Solving Mixture Problems Example continued 3.) SOLVE 6n + 8(144  n) = 144(7.5) 6n + 1152  8n = 1080 Eliminate the parentheses. 1152  2n = 1080 Combine like terms. 2n = 72 Subtract 1152 from both sides. n = 36 Divide both sides by 2. She should use 36 pounds of the $6 per pound candy. She should use 108 pounds of the $8 per pound candy. (144  n) = 144  36 = 108 Continued

Solving Mixture Problems Example continued 4.) INTERPRET Check: Will using 36 pounds of the $6 per pound candy and 108 pounds of the $8 per pound candy yield 144 pounds of candy costing $7.50 per pound? State: She should use 36 pounds of the $6 per pound candy and 108 pounds of the $8 per pound candy. 6(36) + 8(108) = 144(7.5) ? 216 + 864 = 1080 ? 1080 = 1080 ? 

Distance and Rate Two cars are 350km apart and travel towards each other on the same road. One travels 110kph and the other travels 90kph. How long will it take the two cars to meet? Distance = (Rate) (Time) d = rt

Using a Table Rate Time Distance Car 1 110 x 110x Car 2 90 90x

D1 + D2 = Total Distance apart 110x + 90x = 350 200x = 350 x = 1 D1 + D2 = Total Distance apart 110x + 90x = 350 200x = 350 x = 1.75 Hence, the cars will meet in 1¾ hours