Exercises

EXERCISES A baseball team won two out their last four games. In how many different orders could they have two wins and two losses in four games? Order Win-win-loss-loss Win-loss-loss-win Win-loss-win-loss Loss-loss-win-win

Loss-win-loss-win Loss-win-win-loss

TOTAL: 6

James, John, Peter, and Mary were recently elected as the new class officers (president, vice president, secretary, treasurer) of the sophomore class at Summit College. From the following clues, determine which position each holds. a) Mary is younger than the president but older than treasurer. b) James and the secretary are both the same age, and they are the youngest members of the group. c) Peter and the secretary are next-door neighbors. Legend : X = no, O = yes President James John Peter

Mary

Vicepresident

Secretary

Treasurer

It was sale day at Macy’s and everything was 20% less than the regular price. Peter bought a pair of shoes, and using a coupon, got an additional 10% off the discounted price. The price he paid for the shoes was $36. How much did the shoes cost originally?

Let x: original price

x – x (20%) – [x – x(20%)](10%) = $36 x – x(20%) – x(10%) + x(2%) = $36 x(72%) = $36 x = $36/(72%)

x = $50

 Formula of the area of a circle:  𝐴 = 𝜋𝑟 2  𝑟 = radius = 5 cm

 Formula of the area of a rectangle  𝐴 = (𝑙)(𝑤)

 𝑙 = length = 20 cm  𝑤 = 𝑤𝑖𝑑𝑡ℎ = 10 cm

𝐴𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 − 𝐴𝑐𝑖𝑟𝑐𝑙𝑒𝑠 = 𝐴𝑠ℎ𝑎𝑑𝑒𝑑 𝑟𝑒𝑔𝑖𝑜𝑛 4

(𝑙)(𝑤) − (2)(π)(𝑟 2 ) = 𝐴𝑠ℎ𝑎𝑑𝑒𝑑 𝑟𝑒𝑔𝑖𝑜𝑛 4 (20)(10) − (2)(π)(52 ) = 𝐴𝑠ℎ𝑎𝑑𝑒𝑑 𝑟𝑒𝑔𝑖𝑜𝑛 4 10.73 cm2 = 𝐴𝑠ℎ𝑎𝑑𝑒𝑑 𝑟𝑒𝑔𝑖𝑜𝑛

The Draw a Diagram strategy is useful for visualizing relationships and understanding problems. Diagrams help represent a situation, make a problem less abstract, evoke thought, and keep track of your progress. Drawing a diagram is a common strategy in the elementary mathematics curriculum. It can help picture problems involving distances or shapes.

A farmer built a rectangular fence. Each corner of the rectangle had a post. The shorter sides of the fence each had four equally spaced posts, while the longer sides each had six equally spaced posts. How many posts did she use?

4 equally spaced posts

6 equally spaced posts

Total number of posts = 16

Jane leaves New York on a crosscounty car trip at 7 a.m.. She averages 40 mi/h. Alice plans to take exactly the same route, but does not leave until 8 a.m. She averages 50 mi/h. At what time will she pass Jane? (Assume no interruptions would occur along the way).

Jane 40 mi/h 0 mi

7 a.m.

40 mi

80 mi

120 mi

160 mi

200 mi

8 a.m.

9 a.m.

10 a.m.

11 a.m.

12 a.m.

Alice 50 mi/h 0 mi

8 a.m.

50 mi

100 mi

150 mi

200 mi

9 a.m.

10 a.m.

11 a.m.

12 a.m.

There were 5 cars in a race. The blue car was in front of the green car. The yellow car was behind the green car. The red car was between the blue and green cars. The orange car was in front of the blue car. What was the order of the cars from first to last?

ORDER: Yellow, Green, Red, Blue, Orange

The coach brought the 10 players on Robbie's basketball team to the ice cream parlor to celebrate their big win. The players could order an ice cream cone, a soft drink, or both. 7 players had ice cream and 6 players had a soft drink.

Ice cream

7-x

Soft drink

x

(7-x) + x + (6-x) = 10 x=3

6-x

What’s the Number? 1. It is a 3-digit number. 2. The number is even. 3. The tens digit is greater than the number of sides of an octagon. 4. The hundreds digit is greater than the number of sides of a trapezoid. 5. The ones digit is equal to the number of cups in a pint (2 US cups = 1 pint). 6. The sum of the hundreds and ones digits is less than the tens digit. 7. The sum of the three digits is an even number.

1. The number has 3 digits __ __ __ 2. Since the number is odd, the ones place either be 0, 2, 4, 6, 8. 3. Since the tens digit is greater than the number of sides of an octagon, then the tens digit can only be 9. __ 9 __ 4. Since the hundreds digit is greater than the number of sides of a trapezoid, then the hundreds digit can either be 5, 6, 7, 8, or 9.

5. Since the ones digit is equal to the number of cups in a pint, then the ones digit can only be 2. __ 9 2 6. Since the sum of the hundreds and ones digit is less than the tens digit, then the hundreds digit can only be 5 or 6

7. Since the sum of the three digits is even, than the number is 592.

Find the numbers between 10 and 30 that are divisible by 4. If the product of a number and 3 is both less than 20 and divisible by 4, what is the number? Condition 1: 12, 16, 20, 24, 28 Condition 2: product is divisible by 3 and 4 and is less than 20. among the choices above, 12 can only be the number chosen from Condition 1. Therefore the number is 4

Yolanda, Calvin, and Celina had breakfast together. Each chose a different item: a breakfast taco, cereal with milk, and a bowl of fruit. Use the information to match each person with his or her breakfast. ¥ Yolanda sat next to the person who ate the breakfast taco. ¥ Calvin does not like spicy foods and is allergic to dairy products. Answer: 1. Yolanda either chose cereal with milk or a bowl of fruit. 2. Calvin can only eat a bowl of fruit. 3. Due to condition 2, Yolanda chose the cereal with milk. 4. Therefore, Celina chose the breakfast taco.

Ted, Ken, Allyson, and Janie, two married couples, each have a favorite sport: running, swimming, biking, or golf. Given the ff. clues, determine who likes which sport. 1. Ted hates golf. He agrees with Mark Twain that “golf is nothing more than a good walk spoiled.” 2. Ken wouldn’t run around the block if he didn’t have to, and neither would his wife. 3. Each woman’s favorite sport is featured in a triathlon. 4. Allyson bought her husband a new bike for his birthday to use in his favorite sport.

Golf Ted Ken Allyson Janie

Swimming

Biking

Running

1) Use the digits 3, 4, 5, 6 to make 2 addition problems that will have a sum greater than 100. 2) A number is between 300 and 400. If it is divided by 2, the remainder is 1. If it is divided by 4, 6, or 8, the remainder is 3. If it is divided by 10, the remainder is 5. If it is divided by 3, 5, 7, or 9, the remainder is zero. what is the number?

How old am I? Problem 1 Clues: I am less than 20. I am a multiple of 5. I only have 1 digit. Answer: 5

Problem 2: Clues I am between 30 and 40. I am an odd number. Both my digits are the same. Answer: 33

Mr. and Mrs. Turner have four daughters. Each daughter has two brothers. How many children do Mr. and Mrs. Turner have altogether? Explain. Both brothers can be considered as the brothers of all the four daughters. Therefore, Mr. and Mrs. Turner have 6 children.

You have a barrel of water, an 8-quart pail, a 5-quart pail, and an empty barrel. You need to measure 9 quarts of water. Describe how to measure exactly 9 quarts of water using these two pails. Assume you have a large container that will hold as much water as you need. Solution: 1. Fully fill the 8-qt pail. 2. Fully fill the 5-qt pail with the water from the 8-qt pail. 3. Pour the remaining 3-qt of water from the 8-qt pail to the empty barrel. 4. Pour the all the water from the 5-qt pail back to the previously full barrel. 5. Repeat steps 1-4 to more times.

Practice Problems Use guess and check, or make an orderly listing problem strategy, whichever is more appropriate. A carpenter needs to cut a 24-foot piece of wood into two pieces. One piece must be 6 feet longer than the other piece. Find the lengths of the two pieces.

Solution: Guess and check 5 + 11 = 16 6 + 12 = 18 7 + 13 = 20 8 + 14 = 22 9 + 15 = 24

When a teacher divided her students into groups of four, she had three students remaining. When she divided them into groups of five, she had four students remaining. There were fewer than 40 students in the class. How many students could be in the class? No. of candies

Remainder Groups of 5

Groups of 4

39

4

3

38

3

2

37

2

1

36

1

0

35

0

3

Practice Problems Use guess and check, or make an orderly listing problem strategy, whichever is more appropriate. A student opened his book and realized the product of the numbers on the pages facing each other was 215,760. What were the page numbers of the pages facing each other? Solution: 215,760 = 464.50 Use the whole number before and after 464.50 464 x 465 = 215,760

Practice Problems Use guess and check, or make an orderly listing problem strategy, whichever is more appropriate. Diane has ₱1.02 in change consisting of 25¢, 10¢, 5¢, and1¢. She has twice as many 10¢ as 25¢. She has fewer 5¢ than1¢. What is the least number of coins possible? Solution: Guess and check 25c

10c

5c

1c

Total

No. of coins

1

2

5

12

1.02

20

2

4

2

2

1.02

10

2

4

1

7

1.02

14

Although the second row show a fewer set of coins, it does not follow the set of coins wherein the no. of 5c coins is less than the no. of 1c coins.

Practice Problems Use guess and check, or make an orderly listing problem strategy, whichever is more appropriate. There are cars and motorcycles in a campus parking lot. The parking attendant noticed there are a total of 294 tires and a total of 85 cars and motorcycles. How many cars and how many motorcycles are in the lot? Solution: Let n = no. of motorcycles, 85 – n = no. of cars (2n) + [(85 – n)4] = 294 2n – 4n + 340 = 294 Therefore, there are 23 2n = 46 motorcycles and 62 cars n = 23

Practice Problems Use guess and check, or make an orderly listing problem strategy, whichever is more appropriate. A carpenter needs to cut a 24-foot piece of wood into two pieces. One piece must be 6 feet longer than the other piece. Find the lengths of the two pieces. Solution: Let n = length of one piece, n + 6 = length of the other piece n + (n + 6) = 24 Therefore, the first piece of 2n = 18 woos is 9 feet long and the n=9 second piece is 15 feet long.

Practice Problems Use guess and check, or make an orderly listing problem strategy, whichever is more appropriate. Mrs. Emme has some candy. She wants to split it evenly among the students in her class who correctly solve a math problem. If four students solve the problem correctly, then there will be two extra pieces of candy. If seven students solve the problem correctly, then there will be three extra pieces of candy. What is the minimum number of pieces of candy Mrs. Emme can have?

No. of candies

Remainder 7 correct students

4 correct students

7

0

3

8

1

0

9

2

1

10

3

2

Practice Problems Use guess and check, or make an orderly listing problem strategy, whichever is more appropriate.

The sum of three consecutive even numbers is 342. Find the largest of the three numbers. Solution: Let: n = 1st number, n + 2 = 2nd number, n + 4 = 3rd number n + (n + 2) + (n + 4) = 342 Therefore, the 3 3n = 336 consecutive even numbers n = 112 are 112, 114, and 116.

Practice Problems Use guess and check, or make an orderly listing problem strategy, whichever is more appropriate. Melanie gave her mother a bouquet of 24 flowers. The bouquet was made up of roses, carnations, and daisies. There were twice as many daisies as roses. There were 3 times as many carnations as roses. How many of each kind of flower were in the bouquet? Solution: Let: n = no. roses; 2n = no. of daisies; 3n = no, of carnations Therefore, there are 4 n + 2n + 3n = 24 roses, 8 daises, and 3 6n = 24 carnations. n=4

Practice Problems Use guess and check, or make an orderly listing problem strategy, whichever is more appropriate. A phone company charges a $4 monthly service fee and $0.25 for every minute of long-distance call time. If your telephone budget is $9.41, what is the maximum number of minutes of long-distance calls you can make? Solution: $9.41 −$4 $0.25

= 21.64

We only count 21 number of calls.

Practice Problems Use guess and check, or make an orderly listing problem strategy, whichever is more appropriate. You are a magician. Ask an audience member to pick a number, but make sure she doesn’t tell you what the number is. Give her these directions: “Subtract 3 from it. Multiply the result by 6. Add 10. Divide by 2.” Ask her to tell you the result. How would you determine her original number? Solution: [(x – 3)6 + 10 ]/2 = n, x is the original number and n is the resulting number. When determining the original number, apply this expression: [(2n – 10)/6] + 3 = x.