High School Algebra I Unlocked (2016)

Lesson 1.3. Exact Answers, Approximate Answers, and Estimation

Remember our discussion of handing out party invitations in Lesson 1.2? Well, now let’s pretend that you’re throwing a pizza party for you and 30 other lucky people. You create the guest list, send the invitations, and wait for everyone to RSVP. However, many guests aren’t getting back to you (rude!), and you need to start preparing for the party. How would you go about preparing without knowing exactly how many guests will attend? More specifically, how do you determine how much pizza you should order? This is where exact answers, approximate answers, and estimation come into play.

In this scenario, the exact number of people invited is 30. An exact number is a precise value that is absolute and has no uncertainty. For example, 1 inch = 2.54 centimeters.

Let’s say 15 people have responded to your invitation; 12 will attend and 3 will not attend. With this information, you can approximate the number of people who will come to your pizza party by setting up a proportion:

This question requires you to find the expression that is equivalent to 2x² − 6x − 8. Accordingly, you need to use the distributive property to expand out the answer choices.

For (A), you will find the following: 2(x − 4)(x + 1), 2[(x)(x) + (x)(1) + (−4)(x) + (−4)(1), 2(x² + x − 4x − 4), 2(x² − 3x − 4), and 2x² − 6x − 8. Since this choice matches the expression, and only one answer can be correct, the correct answer is (A).

Then cross-multiply and reduce to find the number of attendees:

Therefore, based on the information you have—that 12 of the 15 invited guests who responded will attend—you can determine that approximately 24 individuals will attend the pizza party. An approximate number is not an exact value, but it’s close enough to the exact value to make meaningful calculations. For example, the value of π is 3.14159265.., but we approximate π to 3.14 when using it in the real world. Approximations also involve rounding a number to its nearest place value. For a number that ends with a digit less than 5, round down to the nearest whole number. Alternatively, for a number that ends with a digit greater than or equal to 5, round up to the nearest whole number. For example, 0.33 would be rounded to 0, 10.2 would be rounded to 10, and 158.98 would be rounded to 159.

Now that you know the approximate number of people attending your pizza party, you can figure out how much pizza to order. This is where estimation comes in. You need to estimate the number of pizza slices each person will eat so that you can order enough pizza. You know that everyone will not eat the same amount, so you estimate that they will eat anywhere from 2 to 4 slices of pizza. So, on average, each individual will eat (2 + 4)/2 = 6/2 = 3 pieces of pizza.

Thus, you estimate that 24 people will eat an average of 3 pieces of pizza, and the total number of pizza you need to order will be 24 × 3 = 72 pieces of pizza.

Here is how you may see exact answers and estimation on the ACT.

When .20202 is multiplied by 10⁵ and then subtracted from 66,666, the result is

A. −46,464

B. 464.98

C. 4,646.4

D. 6,464.6

E. 46,464

An estimate is a close guess of the actual value, which is used when exact information is unknown. In this scenario, we estimated the number of pizza slices based on the average slices eaten per person.

Now try another example.

EXAMPLE

There are 30 people invited to a pizza party. If 24 people attend the party, and each person drinks 12 ounces of soda, exactly how many gallons of soda are consumed at the pizza party? (1 gal = 128 oz)

There’s a lot of information here, so attack the question in steps. Your goal is to find the number of gallons of soda consumed at the party.

First, figure out the total number of ounces of soda consumed at the party by multiplying the number of attendees by 12 ounces.

24 × 12 ounces = 288 ounces

Next, set up a proportion to convert 288 ounces to gallons, using the conversion rate given in the question: 1 gallon = 128 ounces.

Now, isolate x and reduce to find the exact number of gallons of soda that was consumed at the party.

So, the 24 people in attendance at the pizza party drank exactly 9/4, or 2.25, gallons of soda.

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However, you will not always be asked to find the exact answer. Let’s take a look at a different example involving approximations.

EXAMPLE

30 people are invited to a pizza party. If 24 people attend the party, and each person drinks 12 ounces of soda, to the nearest whole number, approximately how many gallons of soda are consumed at the pizza party? (1 gal = 128 oz)

Notice any similarities between this question and Example 2? Any differences? If you noticed that the questions look nearly identical, you’re right! But there’s one important difference: In this question, you need to approximate rather than find the exact number.

Repeat the same process as before to find the number of gallons of soda consumed at the party: Find the total number of ounces, set up a proportion between ounces and gallons, and then convert ounces to gallons.

24 × 12 ounces = 288 ounces

As you can see, we get the same answer as we did in the previous example: The 24 attendees at the pizza party drink exactly 9/4, or 2.25, gallons of soda. But since we need to approximate, round 2.25 to the nearest whole number. Since 2.25 ≈ 2, approximately 2 gallons of soda are consumed.

Great work! You have successfully made it through the first phase of your algebra adventure. Before we move on, let’s practice the skills we learned in this chapter. Make sure that you have a firm grasp of these mathematical techniques, as they will come up throughout the remainder of this book.

To solve this ACT math problem, you would start by multiplying .20202 by 10⁵. Regardless of whether you use a calculator, or simply move the decimal point five places to the right, the answer will be 20,202. Next, you need to subtract 20,202 from 66,666 to find that 66,666 − 20,202 = 46,464, which is (E).

However, while we found the exact answer above, we could have estimated and still arrived at the correct answer. After arriving at the value of 20,202, you could estimate that 20,202 ≈ 20,000, 66,666 ≈ 65,000, and the difference between the two is approximately 65,000 − 20,000 ≈ 45,000. The only answer choice that is close to 45,000 is (E).

DRILL

CHAPTER 1 PRACTICE QUESTIONS

Click here to download a PDF of Chapter 1 Practice Questions.

Directions: Complete the following problems as specified by each question. For extra practice, try using an alternative method to solve the problem or check your work.

1. If x = 14 and y = 2, what is the value of 10 − x + 5y ?

2. Which of the following is equivalent to 12(x + 1)(x − 3)² − 48(x + 1)²(x − 3) ?

A) −12(x + 1)(x − 3)(3x + 7)

B) −36(x + 1)(x − 3)

C) 36(x + 1)(x − 3)

D) 12(x + 1)(x − 3)(3x + 7)

3. To raise money for charity, LuAnn is selling tickets to a silent auction for $15 per person and $25 for a pair of tickets. If LuAnn sells 305 single tickets and 122 pairs of tickets, approximately how much money did LuAnn raise from ticket sales?

A) $7,500

B) $7,600

C) $7,700

D) $7,800

4. Simplify the following expression: ƒ³ + ƒ(ƒ − 2)² − 2ƒ² + 5ƒ

5. What is the greatest common factor of the expression 12x³ − 24x² + 6x ?

6. If a = b = 5 and c = 3, what is the value of ?

7. Simplify the expression (b² − 2b + 1)(b³ − 1).

A) b⁵ − 2b⁴ + b³ − b² + 2b + 1

B) b⁵ − 2b⁴ + b³ + b² − 2b − 1

C) b⁵ − 2b⁴ + b³ − b² + 2b − 1

D) b⁵ + 2b⁴ + b³ − b² + 2b − 1

8. What is the result when 0.4444 is multiplied by 10⁶, added to 222, and then divided by 0.5 ?

9. If Mario runs for 30 minutes on each of the 30 days in April, then for how many total seconds did Mario run in April?

10. If Molly walks her puppy Jasper for 20 minutes every morning, 10 minutes every afternoon, and 25 minutes every evening, then, to the nearest hour, approximately how many hours does Molly walk Jasper per week?

SOLUTIONS TO CHAPTER 1 PRACTICE QUESTIONS

1. 13

Here you’re told that x = 14 and y = 2, and you need to find the value of 10 − x + 5y. Substitute x = 14 and y = 2 into the expression 10 − x + 5y, and simplify as follows:

Therefore, when x = 14 and y = 2, the expression 10 − x + 5y = 13.

2. A

This question requires you to identify an expression equivalent to 12(x + 1)(x − 3)² − 48(x + 1)²(x − 3). In order to find the equivalent expression, first identify the common factors among the terms. Both terms in this expression have common factors of 12, (x + 1), and (x − 3), which can be factored using the distributive property:

Next, simplify the inside of the brackets.

After factoring out the common portions of each term, you will find that the expression 12(x + 1)(x − 3)² − 48(x + 1)²(x − 3) = 12(x + 1)(x − 3)(−3x − 7). If you look at the choices, however, this is not an option! Choices (B) and (C) are clearly incorrect, so take a look at (A) and (D). Unlike the expression (−3x − 7), both (A) and (D) have the term (3x + 7) as the final factor. Therefore, you need to alter the expression to make the factor positive:

Thus, the expression 12(x + 1)(x − 3)² − 48(x + 1)²(x − 3) = −12(x + 1)(x −3)(3x + 7), and the correct answer is (A).

3. B

Here you’re told that LuAnn is selling tickets to a silent auction for $15 per person and $25 for a pair of tickets, and that LuAnn sells 305 single tickets and 122 pairs of tickets. Since you need to determine the amount of money LuAnn raised, start by calculating the amount of money raised from single-ticket sales; LuAnn raised $15 × 305 = $4,575 from single tickets. Next, calculate the amount of money raised from paired-ticket sales; LuAnn raised $25 × 122 = $3,050 from paired-ticket sales. Finally, add the single-ticket and paired-ticket income to find that LuAnn raised $4,575 + $3,050 = $7,625. Therefore, LuAnn raised approximately $7,600, and the correct answer is (B).

4. 2ƒ³ − 6ƒ² + 9ƒ

This question requires you to simplify the expression ƒ³+ ƒ(ƒ − 2)² − 2ƒ² + 5ƒ. Use the power of the distributive property to expand out the expression as follows:

ƒ³+ ƒ(ƒ − 2)² − 2ƒ² + 5ƒ =

ƒ³ + ƒ(ƒ − 2)(ƒ − 2) − 2ƒ² + 5ƒ =

ƒ³ + ƒ(ƒ² − 2ƒ − 2ƒ + 4) − 2ƒ² + 5ƒ =

ƒ³ + ƒ(ƒ² − 4ƒ + 4) − 2ƒ² + 5ƒ =

ƒ³ + ƒ³ − 4ƒ² + 4ƒ − 2ƒ² + 5ƒ

Once you’ve expanded out the expression, combine like terms and simplify:

ƒ³+ ƒ³ − 4ƒ² + 4ƒ − 2ƒ² + 5ƒ =

2ƒ³ − 6ƒ² + 9ƒ

5. 6x

Look at each term in the expression and extract the largest common factor. If you consider the coefficients of each term in the expression 12x³ − 24x² + 6x, the greatest common factor is 6, and if you consider the x-values, the greatest common factor is x. Thus, the greatest common factor of 12x³ − 24x² + 6x is 6x.

6. −3.5

The question tells you that a = b = 5 and c = 3, and you need to determine the value of . So, substitute a = 5, b = 5, and c = 3 into the expression of , and simplify as follows:

Therefore, when a = 5, b = 5, and c = 3, the expression = −3.5.

7. C

This question requires you to simplify the expression (b² − 2b + 1)(b³ − 1). Use the distributive property to expand out the expression, paying attention to the sign changes:

Thus, when the original expression (b² − 2b + 1)(b³ − 1) is expanded, the resulting expression will be b⁵ − 2b⁴ + b³ − b² + 2b − 1, or (C).

8. 889,244

Here you need to determine the result when 0.4444 is multiplied by 10⁶, added to 222, and then divided by 0.5. First, tackle the multiplication.

0.4444 × 10⁶ = 444,400

Next, add the result to 222.

444,400 + 222 = 444,622

Finally, divide the result by 0.5.

Thus, the answer is 889,244.

9. 54,000

You’re told that Mario runs for 30 minutes on each of the 30 days in April, and you are asked to determine the number of seconds Mario ran in April. Start by finding the total number of minutes Mario ran in April. Since he ran for 30 minutes per day for 30 days, he ran for 30 minutes × 30 days = 900 minutes.

However, because the question asks you to find the number of seconds Mario ran, you need to convert between minutes and seconds. Thus, set up a proportion and simplify as follows:

Mario runs for 54,000 seconds during the month of April.

10. 6

Here you are asked to find, to the nearest hour, the number of hours that Molly walks her puppy Jasper per week, given that Molly walks Jasper for 20 minutes every morning, 10 minutes every afternoon, and 25 minutes every evening. Since there is a lot going on in this question, tackle it piece by piece.

If Molly walks Jasper for 20 minutes every morning, 10 minutes every afternoon, and 25 minutes every evening, then Molly walks Jasper for a total of 20 + 10 + 25 = 55 minutes each day. Next, you need to determine how many minutes that Molly walks Jasper per week. Since there are seven days in a week, you can determine that Molly walks Jasper for 7 × 55 minutes = 385 minutes per week.

However, the question asks you to determine the number of hours that Molly walks Jasper, so you need to set up a proportion and simplify:

Therefore, Molly walks Jasper for 6.4167 hours per week. Round this answer to the nearest hour. Thus, to the nearest hour, Molly walks Jasper 6 hours each week.

REFLECT

Congratulations on completing Chapter 1!

Here’s what we just covered.

Rate your confidence in your ability to:

•Identify and define the parts of mathematical expressions and equations

1 2 3 4 5

•Solve simple mathematical expressions with variables

1 2 3 4 5

•Apply the theory behind the associative, commutative, and distributive properties to solve mathematical expressions

1 2 3 4 5

•Identify equivalent expressions

1 2 3 4 5

•Compare and contrast exact, approximate, and estimated answers and solve problems related to these concepts

1 2 3 4 5

If you rated any of these topics lower than you’d like, consider reviewing the corresponding lesson before moving on, especially if you found yourself unable to correctly answer one of the related end-of-chapter questions.

CHAPTER 1 KEY POINTS

Make sure you understand the purpose of every question.

If you encounter a wordy question, break it into smaller pieces.

Review your math vocabulary. Questions will be much easier to understand if you understand the terminology.

Understanding mathematical properties will allow you to conquer tricky math questions more efficiently. You should be the most familiar with the associative, commutative, and distributive properties.

An exact number is a precise value that is absolute and has no uncertainty. For example, 1.48 and 2.261 are exact numbers.

An approximate number is not an exact value, but is close enough to the exact value to make meaningful calculations. For example, 1.48 ≈ 1.5 ≈ 2 and 2.261 ≈ 2.26 ≈ 2.3 ≈ 2.

An estimate is a close guess of the actual value that can be used when exact information is unknown.