Algebra 1 Highschool

CHAPTER: What is the first step in solving word problems?

1 Visualize the problem. The first step is to visualize the problem. See if you can picture what is going on.

Take this word problem:

Lucy's rabbit recently had a litter of bunnies and so did Sam's rabbit. Sam's rabbit gave birth to 4 boy and 3 girl bunnies. Lucy's rabbit gave birth to 2 boy and 8 girl bunnies. How many baby rabbits does Lucy have?

Which of the following is NOT an important part of this word problem:

Sam's rabbit gave birth to 4 boy and 3 girl bunnies. The question is only asking about Lucy's rabbits, so Sam's rabbits are irrelevant. Sam's rabbit and its litter are not important to the problem.

Which of the following is the equation to solve the following word problem?

Sam has 4 red balloons and 8 purple balloons. Judy has 12 red balloons and 4 orange balloons. How many red balloons do Sam and Judy have together?

x = 4 + 12 X is the unknown, in this case, the total number of red balloons. To find X, you need to add Sam's 4 red balloons to Judy's 12 red balloons. So the correct equation is X = 4 + 12.

Sam has 4 RED balloons and 8 PURPLE balloons. Judy has 12 RED balloons and 4 ORANGE balloons. How many RED balloons do Sam and Judy have together?

16 In this problem, we just need to add the red balloons together. Sam has 4 red balloons and Judy as 12 red balloons, so the answer is 16 total red balloons.

Sam's rabbit recently had a litter of bunnies and so did Lucy's rabbit. Sam's rabbit gave birth to 4 boy and 3 girl bunnies. Lucy's rabbit gave birth to 2 boy and 8 girl bunnies. How many boy rabbits were born altogether?

6 Sam's rabbit has 4 boys and Lucy's rabbit has 2 boys. 4 + 2 = 6.

Joe has 8 Dalmatians. Sarah has 3 fewer Dalmatians than Joe. How many Dalmatians do they have together?

13 Remember that we must subtract and add to solve this problem.

8 - 3 = 5 Sarah has 5 Dalmatians.

8 + 5 = 13

They have 13 Dalmatians altogether.

Which of the following is an equation to solve this word problem?

Joe has four blue pens and three green pens. Sarah gives Joe four blue pens and two green pens. Sarah originally had 10 blue pens and 12 green pens. How many blue pens does Sarah now have?

x = 10 - 4 To find the number of blue pens Sarah now has, we need to subtract the number of pens she gave Joe (4) from the number she originally had (10).

So our equations is: x = 10-4

Joe has four blue pens and three green pens. Sarah gives Joe four blue pens and two green pens. Sarah originally had 10 blue pens and 12 green pens. How many blue pens does Sarah now have?

6 Sarah had 10 blue pens and gave Joe 4 of them. She now has 6 blue pens.

10 - 4 = 6

What operations are needed to solve the word problem below?

Joe has 8 Dalmatians. Sarah has 3 fewer Dalmatians than Joe. How many Dalmatians do they have together?

Subtraction and addition Visualize the problem. What's going on? We know that we need to find out how many Dalmatians Sarah has, so we will have to subtract because we know she has 3 fewer than Joe. And we need to find out how many Dalmatians Joe and Sarah have altogether. That requires adding.

When solving a word problem, what should you do after visualizing the problem?

Write the equations. After you visualize the problem, write equations that will help you answer the question.

the three steps to solving word problems are

visualize the problem, write the equation, solve the equations

What is the first step in solving a word problem?

Visualize it. The first step is trying to visualize and see the problem in your mind. You might even make some sketches or make some notes to help yourself begin to visualize the problem.

What parts of a word problem should be sketched?

The important parts that will help answer the problem. When you read the problem, look for the most important parts, in this case, the balloons. You should try to sketch the most important parts of the word problem that will actually help you find the answer.

Both Drew and Barry operate baking companies. They are competitors. This coming weekend they are going head to head in the annual bake-off competition at the county fair. Drew is going to bake 15 of his best known pies such as chocolate mousse and coconut crème. Barry is going to also bake 15 of his best known pies such as sweet potato and key lime. How many total pies will be baked between Drew and Barry?

30 Drew is going to bake 15 pies so visualize 15 steaming pies. Barry is also going to bake 15 pies, so visualize 15 more steaming pies. We want to know the total, so we will do some addition here. 15 + 15 = 30.

Which of the following statements is the most important part of this word problem that you should sketch and make note of to help you solve the problem?

Drew and Barry each have a baking company. They are competitors. This coming weekend they are going head-to-head in the annual bake-off competition at the county fair. Drew is going to bake 15 of his best known pies such as chocolate mousse and coconut creme. Barry is also going to bake 15 of his best-known pies such as sweet potato and key lime. How many total pies will be baked between Drew and Barry?

Drew is going to bake 15 pies. This problem is asking you to work out the total number of pies between Drew and Barry. Thus, an important part of the problem to sketch and make note of are the numbers of pies that Drew and Barry were each going to bake. You should sketch and make note of Drew's 15 pies.

What kinds of notes should you take?

Important notes that will help you solve the problem Take down important notes that will help you solve such as what operation is needed and the numbers involved.

Joy takes a car and a boat to get to work. It takes Joy two hours to get to work each morning. If Joy spends 45 minutes on the boat, how long does she spend driving?

1 hour 15 minutes Personalizing word problems involves restating the word problem using terms that you are more familiar with. In the problem above, practice using yourself and a situation that is personalized to you. Subtract 45 minutes from 2 hours to get one hour and 15 minutes.

Which of the following word problems is the properly restated form of this word problem?

Linda brings four apples and Joey brings six apples. How many apples does each friend get if there are five friends?

When shopping for gifts, you see two gift boxes for sale. One contains four watches and the other contains six watches. If there are five friends and you buy both gift boxes, how many watches will each friend receive? Personalize a word problem with more familiar situations. Instead of visualizing Linda, visualize yourself shopping for gifts using the same numbers, four and six, as well as using five friends.

Six lollipops are divided equally among three friends. If each friend now has 10 lollipops, how many lollipops did each friend originally have?

8 6/3 = 2

10 - 2 = 8

Two trains meet each other in Treeport. One train comes from the east two hours away. The other train comes from the west three hours away. If both trains are each 120 miles away from Treeport, how much faster is the faster train traveling?

20 miles per hour faster 120/2 = 60 miles per hour for one train

120/3 = 40 miles per hour for the other train

60 - 40 = 20 miles per hour faster

What important thing do you have to remember when restating word problems?

The restated word problem must be essentially the same problem. One thing you need to make sure of is that your restated problem is asking for the same solution as the original word problem.

Chapter: Which of these is part of the definition of percentage?

A. A part of a whole

B. Per hundred

C. A discount

D. An increase

2 A and B A percentage, or percent, is a part of a whole per hundred, so answers A and B are correct. Percentage literally means 'per hundred;' you can also think of it as part of a whole. 100% is a whole of something.

Order the following percentages in order from least to greatest.

A. 20%

B. 75%

C. 16%

D. 57%

E. 101%

C, A, D, B, E Because percentages can be ordered from lowest to highest based on place number value, we can order the following percentages as follows: 16%, 20%, 57%, 75%, 101%.

Which of these is a way to write percentages?

All of these are acceptable ways to write percents Three ways to write your percentage are with the percentage symbol (%), decimal form, and fraction form.

How would you write eighty-seven percent using the symbol for percent?

87% The percentage symbol is written as (%); therefore 87% is correct.

Which form is this percentage written in?

1.25

Decimal The number 1.25 is written in decimal form.

You go to the store to buy shoes and realize they are marked at 30% off. If the original price of the shoes is $55, what will the discounted price be?

$38.50 You have $40.00. The jeans you want to buy cost $37.99 and sales tax is 8.5%. How much will the jeans cost including tax, and do you have enough money to get them?

You have $40.00. The jeans you want to buy cost $37.99 and sales tax is 8.5%. How much will the jeans cost including tax, and do you have enough money to get them?

$41.22, no To solve this problem, you can either calculate the sales tax and add it or you can add 100% to the 8.5% since your final price will be 108.5% of your cost. Turning the 108.5% into a decimal, you 1.085. Multiplying this with 37.99, you get 41.22. This is over the $40 that you have, so this means, you don't have enough money to buy this pair of jeans.

Convert 53% to a decimal.

0.53 Turning 53% into a decimal, we divide the 53 by 100 to get 0.53.

Convert 247% to a decimal.

2.47 Using the easy method of converting from a perfect to a decimal, we simply shift the decimal point two places to the left to get 2.47.

You go out to dinner and the bill is $19.38. You want to leave the server a 19% tip. How much money will you leave as a tip?

$3.68 First, converting the percentage to a decimal, you get 0.19. Now, you can multiply the amount by this decimal to get 19.38 * 0.19 = 3.68.

Convert 43.6% to a fraction

109/250 43.6/100 does equal 43.6%, but because there is a decimal in the numerator, you need to multiply it by 10 to remove that decimal. For balance, multiply the denominator by 10 as well, resulting in 436/1000, which reduces to 109/250.

What does the word 'percent' mean?

Per 100 Percent means per 100.

Which of the following represents 27% converted to a fraction?

27/100 27% is 27 per 100 so 27/100.

Convert 28.4% to a fraction

71/250 28.4% as a fraction is 28.4/100. Getting rid of the decimal, you get 284/1000. Simplifying, you get 142/500. Simplifying even further, you get 71/250.

Convert 68% to a fraction

17/25 68% as a fraction is 68/100. Simplifying, you get 17/25.

Which of the following is a percent?

31.4% The percentage is the one with the percent symbol (%) after it.

What is 0.6 as a percent?

60% Changing to a percentage means multiplying by 100 or moving the decimal point two spaces to the right and adding a percentage symbol to get 60%.

What is the number 5.14 as a percent?

514% Notice that you are NOT starting with a decimal < 1. You still multiply by 100 and add a percentage symbol, which gives you 514%.

Convert 54% into a decimal.

0.54 Converting 54% into a decimal means you divide by 100, thus moving the decimal space two spaces to the left to get 0.54 and then you leave out the percentage sign.

Convert 7.5% into a decimal.

0.075 Dividing by 100 and leaving out the percentage sign, you get 0.075.

What type of decimal cannot be turned into a fraction?

a decimal that never ends and whose numbers never repeat

What is 7/20 as a decimal?

0.35

What is the mixed number, 12 1/5, as a decimal?

12.2

What is 1.55 as a fraction?

1 11/20

What is 0.4 as a fraction?

2/5

At Al's Pizza, 57% of the orders included pepperoni pizza. At World's Best Pizza, 11 out of 19 orders included pepperoni pizza. Convert 11/19 to a percent and state at which restaurant did pepperoni pizzas make up a greater percentage of the pizzas sold?

57.9%, World's Best Pizza 11 / 19 = .579 (rounded)

.579 X 100= 57.9

.579 = 57.9%

Because 57.9% is greater than 57%, we know that World's Best Pizza sell a greater percentage of pepperoni pizzas.

Convert 4/25 into a percent.

16% 4 / 25 = 0.16

0.16 X 100 = 16

0.16 = 16%

Convert 7/15 to a percent

46.7% 7 /15 = .467

.467 X 100 = 46.7

.467 = 46.7%

At the coffee shop, 86% of customers left a large tip. In contrast, 5 out of 6 customers at the deli left a large tip.

i) What is 5/6 as a percent?

ii) Which restaurant had a greater percentage of customers who left a large tip?

i) 5/6 = 83.3%

ii) the coffee shop had more tips There are two questions to answer:

i) What is 5/6 as a percent? 5/6 as a percent is approximately equal to 83.3%

ii) Which restaurant had a greater percentage of customers who left a large tip? The coffee shop had a higher percentage of customers that left a large tip (86%)

Convert 1/16 to a percent.

6.25% 1 /16 = .0625

.0625 X 100 = 6.25

.0625 = 6.25%

Cassie buys a box of 36 batteries for 15 dollars. Express the rate in its simplest form.

12 batteries/5 dollars The rate is written by writing as a fraction with the 36 in the numerator and the 15 in the denominator. You get 36 / 15 batteries / dollars. Simplifying, you get 12 / 5 batteries / dollars or 12 batteries / 5 dollars.

An aquarium has 14 seals, 22 penguins and 4 sea turtles. What is the ratio of sea turtles to seals? Remember to reduce your ratio to the smallest numbers possible.

2:7 Remember that a ratio is a comparison. In the problem above we are comparing sea turtles to seals. We can write ratios with a colon that stands in for 'to.' So the correct answer is written 4:14, which stands for 4 sea turtles to 14 seals. 4:14 reduces to 2:7.

Which of the following is an example of a unit rate?

15 slices/1 loaf A unit rate is one where the denominator is 1.

A restaurant sells cheeseburgers to chicken sandwiches in a ratio of 7:2. If the restaurant sells 12 chicken sandwiches one day, how many cheeseburgers does it sell?

42 According to the ratio, for every 7 cheeseburgers sold, 2 chicken sandwiches are also sold. So, if 12 chicken sandwiches are sold one day, then the equivalent ratio will be 7 / 2 = x / 12. Solving for x, then gives an answer of x = 7 12 / 2 = 7 6 = 42.

If a car drives at 60 miles per hour for 4 hours, how far will it go?

240 miles Turning this into a mathematical expression, you get 60 miles / hour * 4 hours. Evaluating it, you get 240 miles.

George earns $12 per hour painting houses. If he works for 7 hours one day, how much does he earn?

$84 Remember the steps. First, we read the entire problem, then identify the question. Next, figure out what you know and don't know. Look for key words. In the problem above, the keyword is dollars per hour for 7 hours. We can solve by multiplying.

$12 X 7 = $84

Calvin has twice as many dimes as nickels. If he has a total of $4, how many nickels does he have?

16 x = number of nickels, 2x = number of dimes

0.05x + 0.10(2x) = 4

Substitute the first equation into the second equation:

0.05x + 0.20x = 4

0.25x = 4

x = 16 nickels (and dimes = 32)

Faith buys gifts for 5 friends. If each gift costs $18, how much did she spend in total?

$90 Pay attention to the keyword 'each' which tells us to multiply.

5 * 18 = $90

Connor has seven more nickels than dimes and twice as many quarters as dimes. If he has $6.20, how many nickels does he have?

16 Let's use x for the number of dimes since the numbers of the other coins are based on the number of dimes.

We have seven more nickels than dimes, so that's x + 7.

And we have twice as many quarters as dimes, so 2x for quarters.

Now let's multiply each by its value:

Dimes = 0.10x

Nickels = 0.05(x + 7)

Quarters = 0.25(2x)

Now we can set up our equation:

0.10x + 0.05(x + 7) + 0.25(2x) = 6.20

x = 9

Based on this information, we know that we have 16 nickels, 9 dimes and 18 quarters.

Jennifer has three more dimes than nickels and three times as many quarters as nickels. If she has $7.50, how many quarters does she have?

24 We have three more dimes than nickels, so that's x + 3. And we have three times as many quarters as nickels, so 3x for quarters.

Based on this information we have 24 quarters, 8 nickels, and 11 dimes.

A painting costs $225. If the sale price is $191.25, what is the percent discount?

15% x / 100 = 191.25 / 225

x = 85

So, 191.25 is 85% of the regular price. That means that the painting is discounted 15%. (100 - 85 = 15).

A pair of shoes is on sale for $96. If the shoes are on sale for 20% off, what was the original price?

$120 If the shoes are 20% off that means you pay 80% of the original price. 80% is the same as 80/100.

The proportion looks like this:

80/100 = 96/x

x = 120

Solve for x: 7/x = 28/16

4 A proportion is just two ratios that equal each other. In this case, x=4 because 7/4=28/16.

If it takes a team of three painters to paint a 7-room house in a day, how many painters are required to paint a 14-room house in a day?

6 3/7=x/14

x=6

3/7=6/14

One box of cereal costs $3. How many boxes can you buy with $15?

5 1/3 = x/15.

x=5

1/3=5/15

Solve 8! (factorial)

40,320 A factorial is the product of all the positive integers equal to and less than your number. 8*7*6*5*4*3*2*1=40320

Calculate 4! (factorial)

24 4*3*2*1=24

Solve the following combination:

10C7

120 10C7 = 10!/(7! (10-7)!) = 10!/(7!3!) = 10x9x8x7!/(3x2x1x7!) = 10x9x8/3x2x1 =120

The local bowling team plays in a 7-team league where each team plays every other team 4 times in a season. Using the combination formula, how many different games will be played in a season?

84 There are a total of 7 teams in the league. Each game requires 2 teams. There are, therefore, 7C2 ways of playing each other. This must happen four times since every team must play every other team four times.

So the answer is 4 times 7C2

= 4 x (7 x 6 x 5!)/(2 x 1 x 5!)

= 4 x (7 x 6 / 2) = 84.

Solve the following combination: 5 C 2.

10 A combination is an arrangement of objects where order does not matter. The formula for a combination is nCr = n!/(r!(n-r)!), where n represents the number of items and r represents the number of items being chosen at a time. in this case 5*2=10

Solve the expression 7P2 (P = permutation)

42 ₇P₂ = 7! / (7 - 2)! = 5040 / 120 = 42

or

7 x 6 = 42

Solve the expression 5P4 (P = permutation )

120 5P4 = 5! / (5 - 4)! = 120 / 5! = 120 or 5 x 4 x 3 x 2 = 120.

Jimmy is making multi-flavored ice cream cones by scooping in different flavors one at a time. Jimmy has 6 different flavors but can only put 3 flavors in each cone. The order of the flavors is important to him as it affects how he tastes each ice cream. How many different arrangements of cones can Jimmy make?

120 6P3 = 6!/(6 - 3)! = 720/6 = 120 or 6 x 5 x 4 = 120.

Solve 8! (the factorial of 8)

40,320 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320

Jane is attempting to unlock her locker but has forgotten her locker combination. The lock uses 3 numbers and includes only the numbers 1 to 9. The digits cannot be repeated in the combination. How many possible locker combinations can be formed?

504 locker combinations 9P3 = 9!/(9 - 3)! = 362,880/720 = 504

Another way to look at it is there are 9 choices for the first number in the combination, 8 for the second number, and 7 for the third number: 9 x 8 x 7 = 504 possible locker combinations.

One hose can fill a swimming pool in 12 hours. A different, larger hose can fill the same pool in half the time. If both hoses are used together, how many hours will it take to fill the pool?

4 hours Using the formula, you get 1/12 + 1/6 = 1/t. Solving for t, you get this.

1/12 + 2/12 = 1/t

3/12 = 1/t

3t = 12

t = 4

Adam and Kevin are roommates. Adam can clean their apartment in 45 minutes. Kevin can clean it in 60 minutes. Rounded to the nearest minute, how much time will it take if they work together?

26 minutes Using the formula, you get 1/45 + 1/60 = 1/t. Solving for t, you get this.

4/180 + 3/180 = 1/t

7/180 = 1/t

7t = 180

t = 25.7

Amy, Barry and Carter are building a fence. Together, it takes them 4 hours. If Amy worked alone, it would take her 14 hours. If Carter worked alone, it would take him 16 hours. Rounded to the nearest hour, how long would it take Barry to build the fence alone?

9 hours Using the formula, you get 1/14 + 1/16 + 1/B = 1/4. Solving for B, you get this.

8/112 + 7/112 + 1/B = 28/112

15/112 + 1/B = 28/112

1/B = 13/112

112 = 13B

B = 8.6

Emily can rake her lawn in 5 hours. Andrew can rake the same lawn in 8 hours. Rounded to the nearest hour, how long will the job take if they work together?

3 hours Using the formula, you get 1/5 + 1/8 = 1/t. Now solving for t, you get this.

8/40 + 5/40 = 1/t

13/40 = 1/t

13t = 40

t = 3.08

Farmer John can till his field 1.5 times faster than Farmer Bob. If they work together, they can till the field in 6 hours. How long would it take Farmer John on his own?

10 hours Let's let J = Farmer John and B = Farmer Bob

Farmer John is 1.5 times faster than Farmer Bob, so 1.5 times Farmer John's time equals Farmer Bob's time.

(Think about this for a moment, faster means it takes John less time to complete the job.)

1.5J = B

Using the formula, you get 1/B + 1/J = 1/6. Substituting for B, you get this:

1/1.5J + 1/J = 1/6

Solving for J:

1/1.5J + 1.5/1.5J = 1/6

2.5/1.5J = 1/6

2.5*6 = 1.5J

15 = 1.5J

J = 10

How long will it take a car traveling at 32 mph to go 128 miles?

4 hours If t = d/r, then we have t = 128/32 = 4 hours

A train travels at 60 mph for 5 hours, then slows to 50 mph for 2 hours. How many miles does it travel?

400 miles To find the distance plus distance, we need to tackle this problem in two parts. In the first part, the train travels at a rate of 60 mph for a time of 5 hours. With d = rt, we have d = 60 * 5. That's 300 miles.

In the second part, the train travels at a rate of 50 mph for 2 hours. That's d = 50 * 2. So he went 100 miles in the second part. Now we add the two parts, 300 + 100, to get 400 miles.

Olivia rides her bike at an average speed of 12 mph for 2 hours. How far does she go?

24 miles Let's start by identifying what we know from our formula. Our r, or rate, is 12 miles per hour. Our t, or time, is 2 hours. So d = 12 * 2. Our answer is 24 miles.

A bus travels 165 miles in 3 hours. What is its average speed?

55 mph Let's look at our distance formula: d = rt. Here we know our d is 165 and our t is 3. We want to know r. So we can rewrite the formula as r = d/t. r = 165 / 3 = 55 mph.

A car goes 825 miles on 3 tanks of gas. How far can it go on 7 tanks of gas?

1925 miles Let's just set up two ratios. First, 3 tanks goes 825 miles, so we have 3 tanks/825 miles. Next, there's 7 number of tanks to go x miles.

Let's set these equal to each other and cross multiply.

3x = 825*7

825*7 = 5,775

5,775 /3 is 1925 miles

Chapter Consider the numbers 1/2 and 9. Which of the following statements is completely true regarding the most specific classification of these two numbers?

4 1/2 is a rational number and 9 is a natural number. The most specific classification for the number 1/2 is a rational number as it is not an integer. The number 9 is an integer, but can be more specifically classified as a natural number.

Which of these statements is FALSE regarding the number 0?

0 is the smallest natural number. 0 is a real, rational number. It is also the smallest whole number. It is not a natural number, though.

Which of the numbers in the following set are rational numbers?

500, -15, √2, 1/4, 0.5, -2.50, π

500, -15, 1/4, 0.5, -2.50 Rational numbers include integers, natural numbers and whole numbers. √2 and π are not rational numbers. The rest are.

Which family of numbers begins with the numbers 0, 1, 2, 3, …?

Whole numbers Whole numbers include the counting numbers 1, 2, 3, and so on plus the number 0.

Which is NOT a rational number?

√2 The √2 is NOT a rational number. All the others are rational numbers. A rational number is a number that is made by dividing two integers, meaning it can be written as a fraction, such as 1/3, 0.25 (25/100), 0.666... (2/3), 1000 (1000/1).

Which number line would be best for creating a timeline depicting major events of the 1700s?

Because the 1700s is an entire century of events, the image showing a 100 year period starting at 1700 and going to 1800 is correct. The other models show too large of a span of time or incorrect labeling of the time period.

Graph the following daily high temperatures on a number line.

Monday - 78

Tuesday - 72

Wednesday - 65

Thursday - 80

The graph shows increments of two. The best number line is the first image.

I need more help with this

What numbers are graphed on the number line?

63, 85, 96 The graph shows increments of 5. From left to right, it shows 50, 55, 60, 65, 70....and so on. Based on this information, we know the numbers graphed on the line are 63, 85, and 96.

Which number is graphed on the number line?

65 Remember that you can graph rational numbers on a number line by drawing a line with the appropriate scale and then placing a dot at the correct position on the number line. The number on the line is 65.

Convert the following fraction to a decimal: 9/10

0.9 Remember that the line in a fraction is the same as a division symbol. 9 over 10 is the same as 9 divided by 10. So, one method is to divide the numerator by the denominator. 9 divided by 10 is 0.9. And that's our decimal.

Convert the following decimal to a fraction: 0.07

7/100 0.07 = 7/100

Convert the following fraction to a decimal: 2/3

0.667 Divide 2 by 3

2 /3 = 0.667

Convert the following fraction to a decimal: 35/70

0.5 35 over 70 is the same as 35 divided by 70.

35 / 70 = 0.5

Convert the following decimal to a fraction: 0.25

¼ We have 25 hundredths and we can write it as 25/100.

So 25/100 can be simplified by dividing the numerator and denominator by 25, which gives us 1/4.