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Learning Plan 01

Sections 1.1 – 1.3

Questions 1-5 (Quiz Questions) (Read pages 3-9 in the textbook)

Inductive Reasoning Deductive Reasoning Inductive reasoning is the process of arriving at a general conclusion based on observations of specific example.

Deductive reasoning is the process of proving a specific conclusion from one or more statements. A conclusion that is proved to be true by deduction reasoning is called a theorem.

Example You have a bag of marbles. You remove one marble from the bag and you see that it is blue. You remove the second marble and you see that it is also blue. You remove the third marble and it is still blue. Therefore, you make the conclusion that all the marbles in the bag are blue.

Example You know the rule, that all the numbers ending in 0 or 5 are divisible by 5. You pick a certain number, for example, 45. This number ends with a 5, therefore 45 is divisible by 5.

Counterexample Finding a red marble in the bag of the blue marbles will a counterexample to your

statement that all the marbles are blue.

Question 6 (Quiz Question) (more info on page 5 in the textbook) Identify a pattern in each list of numbers. Then use this pattern to find the next number.

8, 12, 16, 20, 24, … Solution

Each number after the first is obtained by adding 4 to the previous number.

24 + 4 = 28

The next number will be 28.

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Question 7 (Quiz Question) Identify a pattern in each list of numbers. Then use this pattern to find the next number.

4, 12, 36, 108, 324, … Solution

Each number after the first is obtained by multiplying the previous number by 3.

324 ∙ 3 = 972

The next number will be 972.

Question 8 (Quiz Question) Identify a pattern in each list of numbers. Then use this pattern to find the next number.

1

4,1

9,

1

14,

1

19,

1

24, …

Solution Each number after the first is obtained by adding 5 to the previous denominator.

1

24 + 5=

1

29

Question 10 (more info on page 16 in the textbook)

Rounding Numbers to a Particular Place Value To round a number to a particular place value:

1. If the digit in the next place to the right is less then 5, drop that digit and all

other following digits. Use zeros to replace any whole-number places dropped.

2. If the digit in the next place to the right is 5 or greater, add 1 to the digit in the

place to which you are rounding. Drop all other following digits. Use zeros to

replace any whole-number places dropped.

Round 7.4825 to the nearest thousandth. Solution: The thousandths digit is 2. To the right of 2 we have number 5, so we must increase 2 by 1, and drop any digits to the right.

7.4825 ≈ 7.483

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Questions 11, 12 6 (Quiz Question) (more info on page 15 in the textbook)

Rounding Whole Numbers

1. Look at the digit to the right of the digit where the rounding is to occur.

2. If the digit to the right is 5 or greater, add 1 to the digit to be rounded. Replace all digits to the right with zeros. If the digit to the right is less than 5, do not change the digit to be rounded. Replace all digits to the right with zeros.

Round 75,382,895 Solution: To the nearest hundred: 75,382,900 To the nearest thousand: 75,383,000 To the nearest ten thousand: 75,380,000 To the nearest hundred thousand: 75,400,000 To the nearest million: 75,000,000 To the nearest ten million: 80,000,000

Question 13 (more info on page 16 in the textbook)

Rounding Numbers to a Particular Place Value To round a number to a particular place value:

1. If the digit in the next place to the right is less then 5, drop that digit and all

other following digits. Use zeros to replace any whole-number places dropped.

2. If the digit in the next place to the right is 5 or greater, add 1 to the digit in the

place to which you are rounding. Drop all other following digits. Use zeros to

replace any whole-number places dropped.

Round 4.3761 to the nearest thousandth. Solution: The thousandths digit is 6. To the right of 6, we have number 1, so we keep 6, and drop any digits to the right.

4.3761 ≈ 4.376

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Question 14 You run into the grocery store for a few items. You buy a gallon of milk for $4.56, a loaf of bread for $2.72, a package of ground beef for $6.23, a bag of lettuce for $3.29, and your favorite snack mix for $4.45. Estimate to the nearest dollar what your total cost will be. Solution: First, round each amount to the nearest dollar.

$4.56 ≈ $5.00 $2.72 ≈ $3.00 $6.23 ≈ $6.00 $3.29 ≈ $3.00 $4.45 ≈ $4.00

Add all the rounded numbers:

$5.00 + $3.00 + $6.00 + $3.00 + $4.00 = $21

Question 15 Obtain an estimate for the following computation by rounding the numbers so that the resulting arithmetic can easily be perfored by hand or in your head. Then, use the calculator to perform the computation. How reasonable is your estimate when compared to the actual answer?

478 + 365 Solution Round each number to the nearest ten:

478 ≈ 480 365 ≈ 370

Add the rounded numbers:

480 + 370 = 850 Find the actual sum using the calculator:

478 + 365 = 843

Reasonable. The estimate seems a little higher compared to the actual answer.

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Question 16 Estimate the product by rounding the numbers 58.73 × 6.5. Then find the actual product. Solution:

Imagine that you don’t have a calculator, and you must find the approximate product of these two numbers, then you would say that 58.73 is approximately 60, and 6.5 is approximately 7 (according to the rule of

rounding).

Round 58.73 to 60.

Round 6.5 to 7.

60 × 7 = 420

The actual product (use the calculator): 58.73 × 6.5 = 381.745

Question 17 (Quiz Question) The circle graph shows the percentage of 225,540,441 adults who drink caffeinated drinks and the number of cups consumed per day. Without a calculator, estimate the number of adults who drink 3-4 cups a day.

Solution

Round 225,540,441 to 200,000,000.

3-4 cups per day corresponds to 16%. Round 16% to 20% = 0.2

Multiply:

200,000,000 × 0.2 = 40,000,000

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Question 19 (more info on page 30 in the textbook) Nathan wants to buy a sweatshirt. He has a 30% coupon for the in-store purchase. The store charges $50 for the sweatshirt he wants. However, he found the same sweatshirt online for $37 and gets 25% off as a first-time buyer. Which is a better buy? By how much?

Polya’s Four Steps in Problem Solving Step 1 Understand the problem. Step 2 Devise a plan. Step 3 Carry out the plan and solve the problem. Step 4 Look back and check the answer.

Solution:

In-Store Online

The discount is: 0.3 ∙ $50 = $15 The cost is: $50 − $15 = $35

The discount is: 0.25 ∙ $37 = $9.25 The cost is: $37 − $9.25 = $27.75

Online is a better deal. $35 − $27.75 = $7.25

Online is a better deal by $7.25.

Question 20 One more piece of information is given than necessary for solving the problem. Identify the unnecessary piece of information. Then solve the problem. A salesperson receives a weekly salary of $350. In addition, $15 is paid for every item sold in excess of 200 items. How much extra is received from the sale of 212 items? Solution Weekly salary is unnecessary information. 212 − 200 = 12 items sold in excess of 200. 12 × $15 = $180 extra is received.

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Question 21 A storeowner orders 25 calculators that cost $47 each. The storeowner can sell each calculator for $51. The storeowner sold 22 calculators to customers. He had to return 3 calculators that were never sold and pay a $2 charge for each returned calculator. What is the storeowner’s profit? This problem could be solved using different steps. I will show you just one way of solving it.

Expense: 25 ∙ $47 = $1175

Gross Sales: 22 ∙ $51 = $1122

The storeowner returns 3 calculators and receives $45 for each (47 − 2 = 45).

The total received is:

3 ∙ $45 = $135

Total Income: $1122 + $135 = $1257

Total Income – Expense = Profit $1257 − $1175 = $82

Question 22 The members of the Student Activity Councel on your campus are meeting to select three speakers for a month-long event celebrating artists and entartainers. The first names of the choices are John (J), Will (W), Anna (A), and Sofia (S). How many different ways can the three speakers be selelcted? Solution

Keep in mind that the order does not matter. JWA JAS WAS JWS

4 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑤𝑎𝑦𝑠.

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Question 23 (more info on pages 33-34 in the textbook) In how many ways can you receive $0.22? Solution Start with 22 pennies. Then each time take 5 cents and move them to the right and convert them into a nickel, then 2 nickels into a dime, and so on.

Pennies Nickels Dimes

First Way 22 0 0

Second Way 17 1 0

Third Way 12 2 0

Forth Way 12 0 1

Fifth Way 7 3 0

Sixth Way 7 1 1

Seventh Way 2 4 0

Eighth Way 2 2 1

Ninth Way 2 0 2

So, there are nine ways to receive $0.22.

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Question 24 You throw three darts at the board shown. Each dart hits the board and scores 3, 9, or 27. How many different total scores can you make?

Solution Make a list of different scores. Keep in mind that the order does not matter.

3 + 3 + 3 = 9 3 + 3 + 9 = 15 3 + 9 + 9 = 21

3 + 9 + 27 = 39 3 + 3 + 27 = 36

3 + 27 + 27 = 57 9 + 9 + 27 = 45 9 + 9 + 9 = 27

9 + 27 + 27 = 63 27 + 27 + 27 = 81

There are 10 scores.

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Question 25 Five people (A, B, C, D, and E) agreed to share the expenses of a surprise birthday party equally. If A spent $38, B spent $12, C spent $26, D spent $37, and E spent $38, who owes money after the party and how much do they owe? To whom is money owed and how much should they receive? In order to resolve these discrepancies, who should pay how much to whom? Solution

Find the average:

38 + 12 + 26 + 37 + 38

5 𝑓𝑟𝑖𝑒𝑛𝑑𝑠= $30.20

The friends who spent less than $30.20 and owe money to their friends are B and C. Friend B owes: 30.20 − 12 = $18.20 Friends C owes: 30.20 − 26 = $4.20 The money is owed to friends A, D, and E.

Question 26 (Quiz Question) Five people shake hands with each other. How many handshakes are possible? Solution I will show two ways of solving this problem:

First Method Second Method

There are 5 people.

𝑛 = 5

Use the formula: 𝑛(𝑛 − 1)

2

5 ∙ (5 − 1)

2=

5 ∙ 4

2= 10

There are 5 people.

The first one shakes hands with the other 4.

The second one shakes hands with the other 3. The third one shakes hands with the other 2. The forth one shakes hands with the 1 left.

4 + 3 + 2 + 1 = 10

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Question 27 A car rents for $140 per week plus $0.3 per mile. Find the rental cost for a three-week trip of 500 miles. Solution

140 + 140 + 140 + 0.3 ∙ 500 = $570 When it comes to expressions with multiple operations, make sure you know how to use you calculator correctry.

Type the expression above first, and see if you get the same answer.

Question 20 Andrea decided to rollerblade to her mother’s house. Three blocks from her home, one of the wheels on her skate broke, and she had to walk the remaining eight blocks to her mother’s. She could not repair her skate and had to walk all the way back home. How many more blocks did Andrea walk than she skated? Solution

Andrea walked 8 + 11 = 19 blocks. Andrea skated 3 blocks.

19 − 3 = 16 blocks.

Andrea walked 16 more blocks than she skated.

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Quiz Question A store received 400 containers of milk to be sold by February 1. Each container cost the store $0.72 and sold for $1.53. The store signed a contract with the distributor in which the distributor agreed to a $0.50 refund for every container not sold by February 1. If 360 containers were sold by February 1, how much profit did the store make? Solution

$1.53 − 0.72 = $0.81 – the profit for each sold container.

360 ∙ 0.81 = $291.60 – the profit from 360 sold containers.

400 − 360 = 40 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑟𝑠 were returned to the distributor.

40 ∙ (0.72 − 0.50) = $8.80 – the loss for the 40 containers.

$291.60 − $8.80 = $282.80