MPM 1D Name:Unit #3: Solving Equations

Name: __________________________________________________________

Unit Outline:

Date Lesson Title Assignment Completed

3.1 Solving Simple Equations

3.2 Simplifying before Solving

3.3 Solving Multi-Step Equations

3.4 Soling Equations with Fractions

Quiz #1 – Solving Equations (Lessons 3.1 to 3.4)

3.5 Working with Equations and Formulas

3.6 Solving Word Problems

3.7 Solving Ratios and Proportions

3.8 Solving Percent Problems

Quiz #2 – Solving Equations (Lessons 3.5 to 3.8)

3.9 Review of Solving Equations

Unit Test – Solving Equations (Lesson 3.1 to 3.9)

MPM 1D Name:3.1 Solving Simple Equations

Warm Up:Write three more examples of an expression and three more examples of an equation.

Expressions Equations5x – 7

9x – 8 + 2x

4(x – 6)

3x + 2 = 5

3x – 4 = 8x – 9

8(3x – 1) = 32

Compare expressions to equations. State what is similar and what is different.

Fill in the blanks with a number that makes the equation true:

a) 3 + ____ = 8 b) 5 – ____ = 1 c) 15 + _____ = 10

d) –5 + ____ = -7 e) – 3 + ____ = 8 f) 7 – _____ = 13

g) 4 x _____ = 24 h) _____ ÷ -2 = 12 i) 5 ( ____ ) = -30

j ) 12 ____ = -3 k) _____ 4 = -2 l) ____ (-9) = 4

m ) (___ 2) + 7 = 13 n) (____)(9) - 4 = 32 o) 5 ( _____ ) - 6 = 44

Explain your strategy for solving question m) above

MPM 1D Name:Understanding Equations

An equation is a mathematical sentence that uses an equal sign to show that two expressions are equal.

4 + 6 = 10 is true. 2(15) = 30 is true.

An equation is like a scale with the equal sign in the middle:

x + 2 = 5Both sides have the same value. They must always stay in balance.

What value of x will make the scale balance? _______________ Is that the only value? ________

To solve an equation, we must determine the value of the variable that makes the equation true.

Working with the Balance Model

Balance Scales are often used as a physical representation of Algebra Equations.

What happens if we add 3 pounds to the left side?

But, what happens if we add 3 pounds to BOTH sides?

MPM 1D Name:Practice Problems: How can we find out what one triangle is equal to in the problems below?

Reflect: What is the strategy to solving equations? Explain it in your own words.

Solving Equations Using Opposite Operations

Addition is opposite to , and Multiplication is opposite to

Subtraction is opposite to , and Division is opposite to

As the equations get more complicated, solving by inspection becomes very difficult.

Opposite operations always works, no matter how complicated the equation may be.

Let’s practice solving these easy equations using opposite operations.

2 59

8 4 10

99999999999999999999999999999999

88888888888888888888888888888888

MPM 1D Name:Practice Problems: Solve the following equations using opposite operations.

a) a – 8 = 12 b) 5 + k = -10 c) m 9 = 2

c) 8 a = 16 d) 2 b = 20 e) 4 c = 28

f) 15 + x = 22 g) 10 + y = 3 h) b – 12 = 20

i) x – 7 = 15 j) –5 + t = 12 k) –7 + b = -14

l) m) y 6 = - 46 n)

Why are these types of equations called one-step equations?

MPM 1D Name:Multi-Step Opposite Operations

Consider putting on socks and shoes vs. taking socks and shoes off

What do you notice?

When we are is evaluating we always use order of operations ____________________________

When we are is solving with the reserve of order of operations ____________________________

Given the equations below, list the operations that are affecting the variable, then list the operations to undo them.

Solve the equations above.

Getting Dressed Getting Undressed

1. Start with foot 1. Finished

2. Add Sock 2. Untie Laces

3. Add Shoe 3. Subtract Shoe

4. Tie Laces 4. Subtract Sock

5. Finished 5. End with foot

MPM 1D Name:Practice Problems: Solve each of the following equations:

a) 2x - 10 = 26 b) 2x + 3 = 21 c) 3x - 2 = 7

d) -7 = 2x - 21 e) 27 = 4x - 5 f) –x + 9 = 12

Reflect:1. What is different about question d? Does it make a difference when you are solving?

2. Why is equation f considered a two-step equation?

3. If I asked you to solve the equation x2 = 25,

a. What is the inverse operation of squaring a number?

b. What is a solution?

c. Is there a second solution that works?

MPM 1D Name:Assignment 3.1 Solving Simple Equations

1. Look at the algebraic expressions and equations below. Which are expressions? Which are equations? Explain how you know?

a) 5x = 65 b) y + 8 c) 3a – 6 d) z + 3 = 9 e) f) 3q – 5 = 19

2. Solve.a) –3k = 18 b) b – 3 = 12 c) –c = 3 d) 6w – 4 = –22

e) –2g + 3 = –4 f) 5s + 3 = 2 g) 4

x = 6 h) 3d – 5 = –1

3. Solve

a) 14+t=−3

8 b) w−12=−3

5 c) 2 y+23=1

6

4. Find two solutions for each of the following equationsa) x2 = 81 b) y2 = 144 c) m2 = 86

5. Create a two-step equation that can be solved using opposite operations. The solution to your equation must be w = -3.

6. Describe the solution to these equationsa) 3x = 0 b) 0y = 2 c) 0z = 0

7. For each situation below, write an equation you could use to solve each problem. Solve each equationa) Abba bought 15 DVD’s for $255. She paid the same amount for each DVD. How much did Abba pay for each DVD?b) A banquet hall charged $120 for the rental of the hall, plus $14 for each meal served. The total bill for the banquet was $610. How many people attended the banquet?

8. a) One more than three times a number is 28. What is the number?b) Four less than five times a number is 31. What is the number?c) Twice a number increased by seven is 29. What is the number?d) Seventeen added to three times a number is 53. What is the number?

MPM 1D Name:3.2 Simplifying before Solving

Warm UpSimplify.

a) 4x + 7 –3x + 1

b) 3 + 3w –2 – w

c) 5b + 2 + 4b + 6

d) 9 – 7a + 4 + 9a

Simplifya) 3(2m + 1)

b) –6(1 – 3v)

c) –(n + 2)

d) 3(5 + 2g)

Simplifya. (3x – 7) + (x + 9)

b. (2x – 4) – (5x +1)

c. 12x – 3(4x – 3)

d. 3(2x + 5) + 4(x – 8)

Simplify the following expressions, then evaluate for the given values

a. 4w + 5 if w = -3

b. -5(2x – 1) if x = 6

c. 8x – 5 – x + 9 if x = 7

d. 2(4m – 3) – 2(m + 5) if m = -4

MPM 1D Name:Back to the Balance ModelBuild and/or solve the equations.

Problem Visual

2(x+5) = 141)

2f – 12 + 3f = 62)

Practice Problem: Solve the of the following equation; write your steps down as you go

2(3t + 5) – 4(2t – 1) = 6

MPM 1D Name:Mental and Formal Checks

A mental check is a quick way of determining if your answer is correct. Simply substitute your answer back into the original equation and then evaluate.

Practice Problem:

a) Ms. Lindsay solved 5x + 6 = -9 and said that the answer was -3. How could you check to see if she is correct without actually solving it yourself?

b) Solve the following equation 2(3t + 5) – 4(2t – 1) = 6, then perform a mental check to determine if you answer is correct.

MPM 1D Name:A formal check is more mathematical and elegant. It involves following a set of rules agreed upon by all mathematicians to prove (beyond a reasonable doubt) that the answer is in fact correct. Formal checks become much more useful and important in higher grades.

Practice Problem:a) Perform a formal check on the equation 6w – 4 = –22 to determine if w = -3 is the correct answer.

b) Solve the following equation 5(k + 3) – 2(4k + 7) = -5, then perform a formal check.

Formal Check Instructions:1) State what you are checking2) State what the left side of the equation equals3) State what the right side of the equation equals4) Substitute the value5) Simplify each side of the equation SEPARATELY6) Make your concluding statement

MPM 1D Name:Practice Problems: 1. A rectangle has a length of 7x – 4 and a width of 8 – 2x. The perimeter of the rectangle is 38 cm. Determine the length of each of the sides.

2. The figure below has a perimeter of 62 m. Determine the length of side AB. Show your work.

MPM 1D Name:Assignment 3.2 Simplifying before Solving

1. Solve.a) 6x + 3 + 2x = 19 b) 10m – 3m + 8 = 43c) 4a + a + 9 = 44 d) 15 – 3b + b = 3e) 2y + 4 + 3y = 9 f) 7f – 12 + f = 20

2. Solve.a) 4(x – 3) – 3x = – 7 b) 2(a – 8) + 3(a + 6) = 17c) 12 = 4(d + 2) + 3 – 5d + 9 d) 2(3t + 5) – 4(2t – 1) = 6

3. Solve, then perform a formal check.a) 2m + 1 – m = 4 b) 5(k + 3) – 2(4k + 7) = -5

4. The perimeter of the triangle (on right) is 40 cm. Find the value of x.

5. The perimeter of an isosceles triangle is 21 cm. The length of each equal side is triple the length of the base. Find the side lengths of the triangle.

6. Translate each sentence below into an equation and solve for the number described.

a) Three times the sum of a number and four is 45.b) The sum of four times a number and six times a number increased by eleven is 51.c) Four less than three times a number increased by six times the same number is 32.d) Three times a number subtracted from 4 less than seven times a number is 28.

7. Solve the equations

a) b)

Enrichment: Solve the equation

x + 12

83x

MPM 1D Name:3.3 Solving Multi-Step Equations

Warm Up:(a) 3k – 9 = 6 (b) 4 – 2x + 6 – 3x = - 5 (c) 5(x + 4) = 40 – (-5)

Perform a formal check to determine if x = -11 is a solution to the following equations

a) 7x – 9x – 6 = 21 – 5 b) -2x + 9 – x = -20 - 4

MPM 1D Name:Solving Equations with Variables on Both Sides Using the Balance Model

Problem Equation and Solution

1)

2)

3)2x+5 = 3x+1

8)2(3x+1) = x+22

9)5x-2x+8 = 2x+1+2x

List the rules you discovered in this lesson that will help you solve equations when there are variables on both sides?

5

84

MPM 1D Name:Practice Problems:

1) A square and an equilateral triangle are pictured below. The square and the triangle have the same perimeter. What is the value of x?

2) Solve the following equations

a. b.

3) A health club charges nonmembers $2 per day to swim and $5 per day for aerobic classes. Members pay a yearly fee of $200 plus $3 per day for aerobic classes. Write and solve an equation to find the number of days you must use the club to justify a yearly membership.

MPM 1D Name:Assignment 3.3 Solving Multi-Step Equations

 1. Solvea) 3b + 4 = 2b + 6 b) 7p – 18 = 3p – 2c) 2x + 4 = 5x – 5 d) 8g + 3 = g + 10e) 6h – 5 = 2h + 3 f) 4m – 9 = m + 7

2. Solvea) 2 + (4h – 1) = 11 + 2h b) 8 – (2g + 3) = 3g – 5c) 2(d + 6) = 9(d – 1) d) 5(3r – 7) + r = 3(r – 3)

3. Solvea) 4s + 3 – s = –6 b) p – 3 + 2p – 9 = 0c) 5 – (c + 3) = 4 + c d) 3(4d – 7) – 6 = 2(d + 2) – 1

4. Solve, then perform a formal check: 6 – 3(4k + 1) = 5 + (10 – 8k)

5. A regular polygon has equal sides. In the diagram below, the perimeter of the regular hexagon is equal to the perimeter of the equilateral triangle.

3x-2

5x+8

a) Determine the value of x.b) What is the perimeter of the equilateral triangle?

6. Solve the equation

Enrichmenta) Show that the equation has no solution. Why do you think this happens?

b) Show that the equation has an infinite number of solutions. Why do you think this happens?

MPM 1D Name:

3.4 Solving Equations with Fractions

Warm Up:Evaluate

(a)

3

5

4

3

(b)

35

23 (d)

14

45

(b)

26

18

(e)

38

14

(f)

23

49

15

Solve the following equations

(a) 1554 xx (b) 12713 xx

MPM 1D Name:Solving Equations with Fractions

Solve:

If we tried to expand this equation in the first step, we would be left with more fractions. See if you can continue from here…

But what if I told you that it is possible to clear the fraction in the first step!!!But how ???

MPM 1D Name:We want to “get rid” of the denominators!

Step 1: Find the least common denominator for the equation Step 2: Multiply every term in the equation by the least common denominator Step 3: Reduce each term to create a “denominator free” equation Step 4: Solve for the variable using the steps to solve an equation

If there is only 1 fraction in the equation:

1st: Multiply EVERY piece of the equation by the denominator of that fraction.

2nd: The fraction will disappear and you can proceed as usual!

1) 2) 3)

Independent Practice:

1) 2) 3)

MPM 1D Name:

If there are 2 or more fractions in the equation:1st: Find the Least Common Denominator (LCD) of all of the denominators2nd: Multiply EVERY piece of the equation by the LCD

*Remember to put all (binomials in parentheses) before multiplying!3rd: Simplify and solve!

4) 5) 6)

5)

☼ Make sure you distribute and watch out for the negative!

6)

LCD: ___ LCD: ___ LCD: ___

LCD: ___ LCD: ___

MPM 1D Name:Challenge Questions:Follow the four steps to solve the following equations. Show all of your work!

1) 2)

3) 4)

MPM 1D Name:

Assignment 3.4: Solving Equations with Fractions

A. Solve.

(1) 21

4

x

(2) 31

12

y

(3) 5108 n

(4) 31

6

m

(5) 1

32yy

(6) 1

54yy

(7) 67

34

25

nn

(8) ppp

43

43

2

(9) 4

52

3

nn

(10) 5)1(

3)1(

xx

(11) 4)32(

5)3( yy

(12) 4)1(

2)32(

xx

(13) 7

62

31

xx

(14) 3

67

21

xx

(15) 1

325

nn

(16) xxx

23

354

(17) 55

314

kk

(18) 7

241

xx

(19) 1

232

31

xx

(20) 1

72

31

zz

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3.5 Working with Equations and Formulas

The following formula can be used to convert temperatures from degrees Celsius, °C, to degrees Fahrenheit, °F:

51609

CF

.a) Use the formula -7°C to degrees Fahrenheit.

b) Use the formula to find out at what temperature F = C.

Formulas are often used in both math and science, write down as many formulas as you can remember

The formula for determining the surface area of a

cylinder is . Determine the height of a cylinder with surface area of 505 cm2 and a radius of 4 cm.

Solve the following equations a) x + y = 6 b) 3x – y = 10

What problems did you encounter?

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Rearranging Formulas

When working with math, (whether in Chemistry class, Physics class, Business class, etc), you will be working with equations that will need to be rearranged. This makes them easier to work with.

Rearranging equations is a very important skill in mathematics and follows the same rules as solving equations.

Practice Problems: Rearrange each formula to isolate the variable indicated.

a) 2bhA

for h b) 2y + 3x = 30 for y

c) for h d) V = r 2h for r

Standard Form of an EquationSince equations can be rearranged in a number of different ways, mathematicians have agreed upon a set of rules known as standard form of an equation. Equations in standard form must….

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Practice Problems: Rearrange each equation into standard form.

a) b)

c) d)

Challenge Questions: Rearrange in terms of the given variable

a) Ax + By + C = 0 for y b) a2+b2=c2

for b

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Assignment 3.5 Working with Equations and Formulas

1. Rearrange each formula to isolate the variable indicated.a. P = 4s for s b. I = Prt for P

c. 2

bhAfor b d. P = 2l + 2w for l

e. d = st for t f. V = r 2h for h

2. Solve the following equations for y:a. x + y = 10 b. y + 2x = 4 c. –6x = 2y + 3d. 3 + y = 8x e. –y – 5x = -6 f. 10y – 10 = x

3. Rearrange the following equations to standard form.a. 4x + 6y = 8 b. –6x – 7y – 9 = 3 c. x = 8y – 4d. 9y = -5x + 6 e. 7y = 4x f. 4x – 8y + 10z = 2x

g. 5x –7y = 2x –19y – 30 h. j.

4. Solve each formula for the variable indicated.

a. hrV 2 for h

b. rmvC

2

for r

c. 12dNS

for d

d.2

21 gts

for g

e.2

21

dmgm

F for 1m

f. tPA Pr for t

Enriched: Solve each formula for the variable indicated.

a) tPA Pr for P b)

ab= cd for a b)

ab= cd for d

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3.6 Solving Word Problems

Warm Up1. Determine the next two consecutive integers in each case.

(a) -5, -4, -3, _____, _____,

(b) 12,13,14, _____, _____, _____

(c) x, _________, _________, _________

2. Determine the next three consecutive even integers in each case.

(a) -18, -16, -14, _____, _____, _____

(b) 20,22, _____, _____, _____,

(c) x, _________, _________, _________

3. Determine the next three consecutive odd integers in each case.

(a) -13, -11, -9, _____, ______, _____

(b) -5, -3, -1, _____, ______, _____

(c) x, _________, _______, _______

Write down words that mean the following mathematical operations.

Addition

Subtraction

Multiplication

Division

Equals

Translate each of the following sentences into an equation. Do not solve.

a) Ten less than triple a number is twenty-one.

b) The sum of nine times a number and five is one hundred eighty-five.

c) A number divided by six is twenty-one.

d) Double a number plus five is seventy-five.

e) You get ten when subtracting sixteen from twice a number.

f) If a number is tripled and then reduced by nine, the result is sixty-six

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STEPS IN SOLVING WORD PROBLEMS

1. Define the variable that you want to find with a let statement. 2. Create an equation that expresses the information given in the problem’s scenario. 3. Solve your equation using algebraic methods. 4. Consider if your answer is reasonable. 5. Write a concluding statement, be sure to include units.6. Check your answer with the conditions given in the problem.

Practice Problems:

1) Quin was shopping at a used book sale where all books were selling at the same price. He bought six science fiction books and eight mysteries. He also decided to buy a poster for $2.40. In total, Quin spent $8.70. What was the price of a single book?

2) Oberon Cell Phone Company advertises service for 3 cents per minute plus a monthly fee of $29.95. If Parker’s phone bill for October was $38.95, find the number of minutes he used.

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3) Rachael and Sabine belong to different local gyms. Rachael pays $35 per month and a one-time registration fee of $15. Sabine pays only $25 per month but had to pay a $75 registration fee. After how many months will Rachael and Sabine have spent the same amount on their gym memberships?

4) Ella has an older sister and a younger sister. Her older sister is one year more than twice Ella’s age. Ella’s younger sister is three years younger than she is. The sum of their three ages is 26. Find Ella’s age.

5) Find two consecutive integers such that their sum is 89.

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6) Find three consecutive even integers such that the sum of twice the first and three times the third is fourteen more than four times the second.

Challenge: Andy sold watches for $9 and alarm clocks for $5 at a flea market. His total sales were $287. People bought 4 times as many watches as alarm clocks. How many of each did Andy sell? What were the total dollar sales of each?

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Assignment 3.6 Solving Word Problems

1. Together Barry Sullivan and Mitch Ryan sold a total of 300 homes for Regis Realty. Barry sold 9 times as many homes as Mitch. How many did each sell?

2. Wade purchased three videos and one music CD. The CD cost Wade $12.99. If he paid the same amount for each video and spent a total of $42.96, how much did each video cost?

3. At the market, Xiang bought a bunch of bananas for $0.35 per pound and a frozen pizza for $4.99. The total for Xiang’s purchase was $6.04 without tax. How many pounds of bananas did Xiang buy?

4. Yamir went to the store to buy gardening supplies. A bag of dirt was $3.99 and tulips cost 75 cents per bulb. He bought one bag of dirt and some tulip bulbs and spent a total of $12.24 without tax. How many bulbs did Yamir buy?

5. Zoe is comparing two local yoga programs. Yoga-Weigh charges $90 dollars a month and a registration fee of $35. Essence of Yoga charges $80 per month with a $75 registration fee. After how many months will the two schools charge the same amount?

6. Abbey and Blanca are playing games at the arcade in the mall. Abbey has $20 and is playing a game that costs 50 cents per game. Blanca arrived at the arcade with $22 and is playing a game that costs 75 cents per game.

(a) After how many games will the two girls have the same amount of money left?(b) How much money do they have at this point?

7. The length of a rectangular garden is three feet more than twice its width. If the perimeter of the garden is 114 feet, then what is the width of the garden?

8. Find two consecutive integers that have a sum of −67.

9. Find three consecutive even integers such that their sum is 42.

10. Find three consecutive odd integers that have an average of 13.

11. Find two consecutive even integers such that twice the smaller diminished by twenty is equal to the larger.

12. Find three consecutive odd integers such that twice the sum of the first and third exceeds the second by fifteen.

13. Find three consecutive integers such that the sum of twice the second and three times the third is five less than six times the first.

14. Explain why the sum of two consecutive integers must always be odd.

15. The cost of a pen is 3 times the cost of a pencil. The cost of 4 pencils and 3 pens is $9.75. What is the cost of a pencil? What is the cost of a pen?

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3.7 Solving Ratios and Proportions

Warm Up:Ratio – a comparison of two numbers by divison.

Four out of five cars were red. Write this ratio in three different ways:

Rate – a ratio of two measurments with different units.

Mr. Underwood ran 400 yards in 80 seconds. Write this as a rate

Unit Rate – a rate in which the denominator is 1

In example above, write Mr. Underwood’s unit rate

Jill is selling cookie dough. Three tubs cost $22.50. How much will 8 tubs cost?

A basketball player made 12 free throws in 4 games this year. About how many free throws would you expect the player to make in 100 games?

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Proportion – an equation that shows that two ratios are equivalent.

Fred got two out of every three questions right. If there were six questions, Fred got four questions correct.

23 =

46 is a proportion Note: Reciprocal of proportions are also equal

Solving Proportions

Solve each proportion.

a)

x5=4

7 b)

x9=5

6 c)

67=4x d)

95= 6x+1

Applications – Part to Part Problems

Practice Problems:

1. An employee making $28; 000 receives a raise of $1000. All other employees in the company are given proportional raises. How much of a raise would an employee making $32000 receive?

2. A train travels at a steady speed. It covers 15 miles in 20 minutes. How far will it travel in 7 hours, assuming it continues at the same rate?

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3. In the United States, 21 out of every 100 people are under the age of 15. In a town of 20,000 people, how many people would you expect to be under the age of 15? 15 and over? Would you expect these ratios to be equivalent in every town in the United States?

Applications – Part to Total Problems Practice Problems: 1. The ratio of red marbles to blue marbles is 5 to 7. If there are 156 marbles total, how many red marbles are there?

2. Carl, Rani and Katy win a prize of $600. They decide to share the prize in the ratio of their ages. Carl is 15, Rani is 10 and Katy is 5. How much does each of them get?

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Assignment 3.7 Solving Ratios and Proportions

1. The Crayola crayon company can make 2400 crayons in 4 minutes. How many crayons can they make in 15 minutes?

2. A typist can type 120 words in 100 seconds. At that rate, how many seconds would it take her to type 258 words?

3. Mixing 4 ml of red paint and 15 ml of yellow paint makes orange paint. How much red would be needed if you use 100 ml of yellow paint?

4. A car traveled 130 miles at a constant speed moving for two hours along a highway.What is the distance traveled by the car if it was moving for 4.5 hours at the same speed?

5. The ratio of boys to girls in a school is 4 to 3. If there are 195 girls in the school, then how many students are in the school?

6. The ratio of girls to boys is 13:11 in a school. If there are 1968 students, then how many girls are in the school? How many boys are in the school?

7. A school has 3460 students. If the ratio of boys to girls is 31 to 44, how many more girls are there in the school?

8. To make standard concrete, gravel, sand and cement are mixed in the ratio 5:3:1. I wish to make 180 tonnes of concrete. How much gravel, sand and cement must I purchase?

9. The recommended disinfectant to water ratio is 1: 20. How many mL of concentrated disinfectant are required to make a 9 L bucket of mixture?

10. Joe and Bob share the cost of a video game in the ratio 3:7. a. What fraction does each pay?b. If the game costs $35, how much does each pay?c. If Joe pays $12, how much does Bob pay?d. If Bob pays $42, what is the price of the video game?

11. A student council collects aluminum pop tabs to raise money to purchase a wheel chair. A company buys the pop tabs for $0.88 per kilogram. If 1267 pop tabs have a mass of one pound, how many pop tabs are needed to purchase a wheel chair worth $1500? (Hint 1 kilogram = 2.2 pounds)

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3.8 Solving Percent Problems

Warm Up:What does the word percent mean? When is percent used outside of math class?

Complete the table below: In her math class, Samantha’s two most recent quiz grades are 29/35 and 22/26. Which quiz grade represents the better score?

Estimating percent is very helpful when shopping.

a) Estimate the amount that you should tip on a bill of $45.98.

b) Estimate the amount of tax on a purchase of $134.50.

REDUCED FRACTION DECIMAL PERCENT

155%

0.06

43.2%

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Percent Review

A percent is a ratio between two quantities in which the second quantity is always 100.

84% means 84100 or 0.84.

You need to have a solid working knowledge of percents, not only for success in this course, but also because percents are used in so many areas in our lives outside of school, such as sports statistics, taxes, wage increases, and sales commissions.

Because finding a percent involves work with equivalent fractions, many percent questions are easily handled by solving a proportion of the following type:

parttotal

= percent100

isof

=%100

Practice Problems: Answer each of the following questions by setting up an appropriate proportion and solving for the unknown. Round answers to the nearest tenth, where appropriate.

(a) What number is 18% of 200? (b) 42 is what percent of 98?

(c) 6% of 240,000 is what number? (d) 12% of what number is 1044?

Note: It may be helpful to think of the ratio “part/total” as “is/of.”

The whole is often preceded by the word “of” in the problem.

The part (or percentage) often precedes the word “is” in the problem.

MPM 1D Name:

Percent Word Problems: Answer each of the following exercises by setting up and solving an appropriate proportion.

1) Tanisha is about to sell her house for $315,000. Her real estate broker will expect to receive a 6% commission for all of her hard work in finding Tanisha a buyer for her house. How much money will the real estate broker expect to be paid once Tanisha’s home is sold?

2) Quinn decided to raise all of the prices in his store by 4%. What is the new price of an $8.25 item after the 4% increase?

3) Discount Dave’s is having an end of summer clearance sale. All items are 60% off. What is the sale price of a flat screen TV normally priced at $3499? What is the final cost after 13% sales tax is added on?

Bonus: Sheila claims that if you increase a number by 10% and then decrease the result by 10%, you are right back where you started. Show that Sheila is incorrect by using a specific numerical example.

MPM 1D Name:

Assignment 3.8 Solving Percent Problems

1. 15% of what number is 24? 2. What percent of 300 is 0.75?

2. What is 30% of 95? 4. % of what number is 150?5. What percent of 75 is 375? 6. What is 0.08% of 3200?

7. What is % of 5? 8. 2.5% of what is number 12.5?

9. David has $425 worth of items in his shopping cart. He receives a discount of 30% for the items at the checkout. If he then has to pay an 13% sales tax, what is the final amount that David paid?

10. During a recent clothing sale, a department store offered a discount of 30% off any jacket. Maria paid $45.50 for a jacket after the discount. What was the original price of the jacket?

11. At the Fine Leather Emporium, a $500 leather jacket is on sale for $350. What is the percent of the discount?

12. A math test was passed by 88% of those students who took the test from a certain school. If 792 students passed the test, then how many students did not pass the test?

13. A survey of teenagers from a certain town ages 14 to 18 was taken to see how typical it is for a teenager from that town to carry a cell phone. 88 of the teens surveyed responded that they do carry a cell phone, which represented 80% of the total number of teens surveyed. How many teens were surveyed?

14. After eating a meal at a restaurant, it is appropriate to leave a 15-20% tip for the waiter or waitress. If your total bill at a restaurant comes to $118.00, what should your tip be if you decide to leave a (a) 15%? And (b) 20%?

15. Tammy claims that if you increase a number by 10% and then increase the result by 10%, then you have increased the original number by 20%. Show Tammy that she is incorrect by using a specific numerical example.

16. Margie has 8.5% of her weekly paycheck deposited into her retirement account. This week she had $47.55 deposited into her account. How much was her paycheck

17. Corbett Sprockets Inc. increased its profits by 18% over the past year. If profits last year were $1,400,800, what are its profits this year?

18. Jason works at a local men’s store and is in charge of pricing new items before they are put on the rack. He has a shipment of men’s suits that will be marked up 35%. If each suit costs the store $275.00, what will the selling price be after the mark-up?

20. At the Miracle Plastic Company they had a daily production run of 220,000 units. 1760 of those units were defective. What is the percentage of those units that were defective?

MPM 1D Name:

Answers

Assignment 3.1 1. Expressions: b, c, e

I know they are expressions because they do not have an equals sign.Equations: a, d, fI know they are equations because they each contain an equal sign.

2. a) 6 b) 15 c) 3 d) 3

e)

72 f)

15

g) 24 h)

43

3. a) -5/8 b) -1/10 c) -1/44. a) 9 and (-9) b) 12 and (-12) c) approx. 9.3 and (-9.3)5. answers may vary6. a) x must equal zero b) there is no solution to this equation c) z can be any number7. a) 15n = 255; n = 17; she paid $17 for each DVD.b) 120 + 14n = 610; n = 35; 35 people attended the banquet.8. a) 3n + 1 = 28; n = 9; the number is nine.

b) 5n – 4 = 31; n = 7; the number is seven.c) 2n + 7 = 29; n = 11; the number is eleven.d) 17 + 3n = 53; n = 12; the number is twelve.

Assignment 3.2 1. a) 2 b) 5 c) 7 d) 6 e) 1 f) 4 2. a) 5 b) 3 c) 8 d) 4 3. a) 3 b) 24. 5 cm 5. 3 cm, 9 cm, 9 cm6 a) 3(n+4) = 45; n= 11 b) 4n+6n+11 = 51; n= 4 c)3n -4 + 6n = 32; n = 4d) (7n -4) – 3n = 28; n= 8

7. a) b) x = 3 or x = -3Enrichment: x = 5 or x = -7 Assignment 3.3

 1. a) 2 b) 4 c) 3 d) 1 e) 2 f)163

2. a) 5 b) 2 c) 3 d) 2 3. a) 3 b) 4 c) 1 d) 34. k = -35. a) x = 12 b) P = 204 units

6.

Assignment 3.4A: (1) x=2, (2) y=4,(3) n=4,(4) m=-2 (5) y=-6,(6) y=20,(7)n=-1,(8)p=1,(9)n=15,(10) x=-4,(11) y=-2,(12) x=1,(13) x=14,(14) x=4,(15) n=-9,(16) x=-2,(17) k=5,(18) x=-9,(19) x=2,(20) z=2

MPM 1D Name:

B: (1) hours (2) 20 minutes

Assignment 3.5

1. a) 4Ps

b)IPrt

c)

2Abh

d) 2

Pl w e)

dts

f) 2

Vhr

2. a. y = 10 – x b. y = 4 – 2x c. y = -3x - 23

d. y = 8x – 3 e. y = 6 – 5x f. y = 101

x + 13. a. 4x + 6y – 8 = 0 b. –6x – 7y – 12 = 0 c. x – 8y + 4 = 0 d. 5x + 9y – 6 = 0

e. –4x + 7y = 0 f. 2x – 8y + 10z = 0 h. 3x + 4y – 20 = 0 i. 8x + 3y + 6 = 0

4. a. 2rVh

b. cmvr

2

c. N

sd12

d.

2

2tsg

e. 2

2

1 gmFdm

f. PrPAt

Assignment 3.61. Mitch – 30 houses and Barry 270 houses 2. $9.99 3. 3 pounds4. 11 bulbs 5. 4 months 6. a) 8 games b) $16 left 7. 18 feet8. -34 and -33 9. 12, 14, and 16 10. 11, 13, and 15 11. 22 and 2412. 3, 5 and 7 13. 13, 14 and 15 14. 2n + 115. $0.75 pencil and $2.25 for a pen

Assignment 3.7 1. 9000 crayons 2. 215 seconds 3. 26.67 ml of red 4. 292.5 miles5. 455 students 6. 1066 girls and 902 boys 7. 600 more girls8. 100 tonnes of gravel, 60 tonnes of sand and 20 tonnes of cement

9.

37 ml of disinfectant and

8 47 ml of water

10. a) Joe pays

310 and Bob pays

710 b) Joe pays $10.50 and Bob pays $24.50

c) Bob would pay $28.00 d) Video game costs $60.0011. 4 751 123 pop tabs

Assignment 3.81. 160 2. 0.25% 3. 28.5 4. 450

5. 500% 6. 2.56 7. or 0.025 8. 5009. $336.18 10. $65.00 11. 30% discount 12. 108 students13. 110 teens 14.a) $17.70 b) $23.60 15. Answers may vary 16. $559.4117. $1,652,944 18. $371.25 19. 0.8% defective