ACT Math Lesson 1: Pre-Algebra

Welcome to Elite Academy’s intensive ACT Math course. Over the next 8 lessons, you’ll master all the skills and strategies required to conquer the ACT math section.

Today we'll first go over the components and strategy of the exam. We'll then move on to looking at some important pre-algebraic concepts.  

The Hook: Proportional Mullets

Intro

 

Each lesson will begin with a (hopefully) fun hook that will get us thinking about a key ACT math concept in a more creative and out-of-the-box way than the exam itself might normally trigger. 

Click Next to get started. 

Proportional Mullets

Today we’ll cut out an understanding of proportions by looking at and thinking about fantastic, glorious mullets: 

mullet

Our question today: Which of the above is the truer mullet and why?

If you don’t already know, a mullet is a hairstyle in which the hair is cut short at the front and sides and left long in back. It can be better understood by the magic phrase: 

Business in the front, and party in the back. 

In mathematical terms, we can quantify a mullet’s quality by understanding a simple ratio. 

A mullet’s quality can be measured by the relationship between the “party in the back” to the “business in the front.” The higher the ratio, the more splendid the mullet. 

The hairs in the front of Mullet A are 3.5 inches long, while the hairs in the back are 9 inches. 

The hairs in the front of Mullet B are 1.5 inches long, whereas the hairs in the back are 7.5 inches. 

Which is the more magnificent mullet, A or B? 

Use these measurements to create ratios and justify your answer. 

Overview to ACT Math Section

Test Basics

The ACT Math test consists of  questions that must be answered within a -minute time limit. That leaves only  minute per question, with many questions involving extensive reading and calculation. 

All of the math questions are -choice, questions.  

ACT Math Section Distribution

These 60 questions draw from six areas of math that you should be familiar with already:

  • pre-algebra
  • elementary algebra
  • intermediate algebra
  • coordinate geometry
  • plane geometry
  • trigonometry. 

Because the ACT is a very standardized test, the distribution of questions across the different math skills is quite consistent.

For fun, can you match each subject with its most likely number of questions. 

  • Pre-algebra
    14
  • Elementary Algebra
    10
  • Intermediate Algebra
    9
  • Coordinate Geometry
    9
  • Plane Geometry
    14
  • Trigonometry
    4

Calculator Warning

You may use a calculator for the entirety of the ACT Math exam.

The problems are designed so that you should not need a calculator, though. This is an important fact.

In other words, the numbers work out nicely. You will not, for example, have to convert (253/457) to a decimal or evaluate sin (34°). 

If you do choose to use a calculator, which most students do, your first step should be to make sure that it is legal. The most common calculators used by Elite students are the TI‑83, the TI‑84, and the TI‑89.

The first two are legal, but the third is not. If you have an uncommon calculator, it is a good idea to look on act.org to make sure it is legal. 

Scoring & Organization

You will receive a score from 1 to 36 for the math section as a whole and three sub-scores from 1 to 18, one for categories of Pre-algebra and Elementary Algebra, one for categories of Intermediate Algebra and Coordinate Geometry, and one for categories Plane Geometry and Trigonometry.

Monitoring how you performed on each Subscore category will help you zero in on those areas you need to focus additional practice.

On actual test day, though, you should not worry too much about your final subscores. Just focus on getting as many questions right as you can, regardless of the categories they are in. 

 

 

Scoring

You will receive a score from  to  for the math section as a whole and three sub-scores from 1 to 18, one for categories of Pre-algebra and Elementary Algebra, one for categories of Intermediate Algebra and Coordinate Geometry, and one for categories Plane Geometry and Trigonometry.

Monitoring how you performed on each Subscore category will help you focus in on those areas you need to focus additional practice.

On actual test day, though, you worry too much about your final subscores. Just focus on getting as many questions right as you can, regardless of the categories they are in. 

 

Test Organization

The questions do not appear in any easily decipherable order.

Unlike some other standardized tests, they do not go from easy to hard, and they do not go by category. Since your goal is to get the greatest possible number of questions right, you should not worry about going out of order or skipping questions that seem too hard.

Remember time is limited: If a quick glance at a question tells you that it isn’t for you (maybe because it looks hard or because it just has a lot of reading), don’t hesitate to move on to the next one.

Hopefully, you’ll be able to go back and properly attack it after working through all the easier questions. But don't make one difficult question cost you the potential to answer three or four other questions correctly later on.  

 

  • I should never move on from a difficult question on the ACT-- I need to be perfect!
  • Math questions on the ACT are divided by category

Strategies

Let's take a pause for self-reflection. 

Reflect on how you approached the math section on your most recent ACT exam. 

Were there any special strategies you employed on the past test? What worked and didn't work?  

Following your reflection, we’ll briefly enumerate one basic approach to the section that has worked for Elite students in the past. 

Strategy

  • There is no guessing penalty on the SAT
  • We recommend that you bubble in the answers immediately

Test Strategy Hypotheticals

  • If you have solved a question and are certain it’s correct,
    cross out the problem number to let yourself know not to return to that question.
  • If you have determined an answer for a question but are unsure whether it is correct,
    write down the answer with a question mark next to it. If time permits, you can return to the problem and try again, but if you run out of time, you at least have an answer.
  • If you are all but certain that you cannot solve a problem
    write down your best guess and underline it.
  • When time is about to run out,
    fill in answers for any questions you have not done. It does not matter whether you have even read the questions; never omit on the ACT. Your score is based only on the number of questions right, and there is no deduction for incorrect answers.

Preference

In addition to categorizing questions by the math subject by math subject, we can also break them down in another useful way: 

Basic math problems: 

These questions are straightforward math questions with very few words. Asking you to calculate the remainder when 85 is divided by 7 or to factor a polynomial are examples of such questions. 

Basic math problems “in settings”: 

By “in settings,” the ACT generally means word problems. An example of such a problem might involve calculating the floor area of a rectangular living room. In other words, the problem is a basic rectangle problem but in a specific, real-world context. Not all problems in this category will be quite that simple, but they are not too much more difficult. 

Very challenging problems: 

These problems are straightforward math questions with very few words, but they are harder than the questions in the first category. For example, a question in the first category might ask you the area of a rectangle while a problem in this category might ask you to find the area of a more complicated triangle.

Very challenging problems “in settings”: 

In case you haven’t guessed by now, these are the harder word problems that require complex calculations within a specific, real-world context. 

Question sets: 

In these situations a number of questions, usually from 2 to 4, all relate to a single diagram or situation. Individual questions within this category may also fall into one of the categories above. 

Just curious, do you struggle more in problems with or without many words?

Lesson Complete!

Awesome, awesome work. You clearly know the nature of the Math ACT and how to attack it.

Next, we'll start exploring the world of Pre-Algebra.

If you want to take a break and come back to finish the next lesson later, that's ok!  

 

Ratios, Proportions, and Percentages

Intro

It's time to begin our first proper ACT mathematics lesson.

Today, we will work though the essentials of Ratios, Proportions, and Percentages in fast, but comprehensive fashion. 

Please make sure you're working with pen and paper as you encounter questions, in order to practice the skills you'll need on the exam. 

Ratios

A ratio is quite simply a relationship between two quantities. 

A ratio can be expressed in different forms, including the following three below: 

a/b

a : b

the ratio of a to b 

These three expressions essentially express the same relationship between the two quantities.

The fractional expression a/b is especially common on the ACT. 

A ratio not only gives the relationship between two numbers but also the relationship between any one of the two numbers and the sum of the two numbers.

For example, if the ratio of girls to boys in one class is 3:4, then the ratio of girls to the total number of students in the class is 3:7, simply adding 3 to 4 to get 7; similarly, the ratio of boys to the total number of students in the class is 4:7.

The derivation is as follows: 

Ratio Test Example

Not only does a ratio between two numbers tell us about their proportional relationship, but it can also help us determine a specific quantity of one of the variables when we're given information about the total quantity or the other variable. Let’s apply this point to the following question from our first exam. Try to solve and then review the solution. 

Janelle cut a board 30 feet long into 2 pieces. The ratio of the lengths of the 2 pieces is 2:3. What is the length, to the nearest foot, of the shorter piece? 

  • 5
  • 6
  • 12
  • 15
  • 18

Ratio Practice Example

Now, let’s have you complete a practice question by applying the same principle 

In a mixture of walnuts and Brazil nuts, the ratio, by weight, of walnuts to Brazil nuts is 4 to 3. If the mixture weighs 42 ounces and contains only these two types of nuts, how many ounces of walnuts are in the mixture? 

  • 12
  • 15
  • 18
  • 21
  • 24

Ratio Practice Example II

Basically, you can think of ratios as fractions. To find the ratio of a:b, we can simply divide number a by number b, and then simplify the fraction by removing any common multiple. 

This method works backwards, as well. Given the ratio of a:b, we can list all possible values for a and b, as long as they can be simplified into the given ratio. 

Consider the problem below. 

Balls are to be removed from an urn that contains 20 black balls and 20 white balls. What is the least number of balls that must be removed so that the ratio of the number of black balls to the number of white balls left in the urn is 9 to 2 ? 

 

  • 7
  • 11
  • 15
  • 18
  • 22

Ratios Reflection

How's your ability and confidence in handling ratio problems?

Proportions Practice

A proportion is simply the name we give to an expression of two equal ratios. On the ACT, you'll need to be comfortable with 2 kinds of proportions: Direct and Inverse. 

Let's first look at Direct:

Direct proportions

Two variables, x and y, are said to be directly proportional when the ratio of x:y is constant. So no matter by how much x or y changes, the other variable must change at the same rate, which means the ratio of x to y always stays the same, which means x1/y1=x2/y2

If x is directly proportional to y and x=3 when y=5, what is the value of x when y=15. 

  • 9
  • 13
  • 15
  • 21
  • 25

Proportions Practice

Some ACT word problems will not explicitly state the words “directly proportional,” so we will sometimes need to identify the hidden direct proportion based on the nature and wording of the problem.

Take a look at the example below. 

On a map of a city, every 1/2 inch represents 8 miles. If the length of a certain street is 6 miles, what is the length of the street on the map ?

 

  • 3/16 inch
  • 1/4 inch
  • 5/16 inch
  • 3/8 inch
  • 716 inch

Proportions Practice

In an image of plant cells taken from a microscope at a particular resolution, 1 centimeter represents 5 micrometers. How many centimeters on the image represents 0.5 micrometers ? 

 

  • 0.001
  • 0.02
  • 0.05
  • 0.1
  • 0.3

Inversely Proportional

Inversely proportional

In addition to direct proportions, you will encounter another type of proportion, inverse proportion. For direct proportions, two values change at the same rate; if x increases, then y must increase.

In direct contrast, with inversely proportional types, when x increases, y must decrease, which means their product is a constant.

So, two variables, x and y, are said to be inversely proportional when the product xy is constant, which can be expressed as: 

x1y1=x2y2 

If x and y are inversely proportional and x=12 when y=4, when y=6 the value of x is  ? 

Percentages

We’re all familiar with the general concept of percentages, but how does it relate to ratios?

Quite simply, a percentage is a number or ratio written as a fraction of 100.

Let’s revisit the example of boys and girls in one class at the beginning of this lesson. In a certain class, there are 9 boys and 12 girls, so the ratio of boys to girls is simply calculated as 9/12=3/4.

Let’s say a question asks you to calculate the percentage of boys to girls. With your knowledge of ratios, the answer requires only a couple more steps:

Thus, the number of boys is 75% of the number of girls.

 

Previous examples show one basic way of solving the percentage of two values.

If you are asked to solve a question that asks, “value A is what percent of value B?” then you can simply find the ratio AB, convert into decimals, and then multiply by 100%.

 

Sometimes, a question will supply you with a number and a percentage and ask you to find the resulting number based on the relationship between the given number and the percentage.

 

For example: 8 is 20% of what number?

Based on the definition of percentage, we know that 8x=20%.

So, you can solve the equation by multiplying x on both sides to get

 

Percentage ?

Let’s look at a percentage example from our most recent ACT exam.

If 40% of a given number is 8, then what is 15% of the given number? 

  • A.1.2
  • B.1.8
  • C.3.0
  • D.5.0
  • E.6.5

Percentage Change

ACT Math will ask you to calculate percentage changes based on an original value and a new value, which has either increased or decreased from the original value.

With the basic definition of percentage in mind, let’s study two formulas related to percentage increase and decrease. Here are the formulas: 

Let’s apply the formulas to an example: 

 

Last week the price of Costco gasoline was $2.55 per gallon, and this week the price of the gasoline is $ 2.75 per gallon. What is the percent increase?

  • 6
  • 6.3
  • 6.9
  • 7.8
  • 8.5

Percentage Change Variation

Sometimes, the ACT may ask you to calculate the new value of a variable given the old value and a percentage change. This type of problem is a variation on the percentage change formulas that we've just been practicing. 

To calculate the new value of a variable based on a percentage change to the old value, use this formula: 

New Value = ( 1 ± Percentage Change ) × Old Value

For the above calculation, you will use the minus sign when there is a percentage decrease and a plus sign when it is percentage decrease. 

You should have no problem using algebraic operations to derive this formula based on the percentage formulas introduced previously.

Try it for yourself. 

In a department store, a $100 dress is marked "25% Off." How many dollars is the sale price of the dress? 

  • 25
  • 50
  • 66
  • 75
  • 80

Inverted Percentage Change

This formula works both ways as long as you are given one of two values and a percentage change. 

Try a related inverted problem: 

In a department store, a dress is marked "25% Off," and the sale price is $75. The original price of the dress is dollars 

Inverted Percentage Change

Let's try one markup variation: 

Facing the increasing costs of maintaining the property, a landlord decides to increase the rent by 10%. If the current rent is $2,000 per month for the tenant, the new rent cost for the tenant is dollars.

The extra amount you need to pay is  dollars. 

Consecutive Discounts

We have learned how to calculate the new value given the percentage change of the old value. 

Now, let’s cover a slightly more difficult type of percentage change question, which is called consecutive discount. As the term implies, this type of question involves multiple percentage changes of one value. 

Let’s say you want to buy an item, and it has been marked 20% off. However, it’s your lucky day and you also have a coupon, which gives you another 10% off on top of the 20%. 

If you want to calculate the final purchase price for the item, you can just multiply the new value by the consecutive discounts (1-20%)(1-10%). Try the example below: 

In a department store, a $100 dress is marked “25% Off." In addition, you use a 20% off coupon to buy the dress. What is the sale price of the dress?

  • 40
  • 45
  • 50
  • 55
  • 60

Basic Operations

Basic Operations are the old chestnuts : Addition, Subtraction, Multiplication, and Division.

You will not see basic operations questions such as “What is 1008/14-1008/36 equal to?” on the ACT. Instead, you will need to determine which particular operations will be needed to solve a question by comprehending the relationships between different values in the question.

Once you establish these relationships, you use your knowledge of grade-school math to quickly produce the correct answer. Just be careful to get your relationships clear and avoid silly, sloppy mistakes in calculations. 

Often, these questions will involve multiple computations, so even though they are simple at the level of understanding, they can lead careless students astray. 

Vehicle A averages 14 miles per gallon of gasoline, and Vehicle B averages 36 miles per gallon of gasoline. At these rates, Vehicle A needs  more gallons of gasoline than Vehicle B to make a 1008-mile trip.

Lesson Complete!

Awesome, awesome work. You worked though a number of essential concepts and problems.

Next, you'll perform some problem sets that will reinforce the lesson. 

If you want to take a break and come back to finish the problem sets, that's ok!  

 

Problem Set

Instructions

Each lesson, you will encounter a series of practice problems. These problems will reinforce and build upon the lesson that you worked through. 

Make sure you're in a comfortable, quiet space, ready to focus. 

 

Problem Set 1-5

 

1.A coffee maker from company 𝐴 makes 6 liters of coffee per hour, and a coffee maker from company 𝐵 makes 4 liters of coffee per hour. If Adam makes coffee with two coffee makers from company 𝐴 for two hours, how many hours does Beth need to make the same amount of coffee with three coffee makers from company 𝐵?

 

2.What is the least common multiple of 15, 20, and 25?

 

3. |2|1−2|−3|3−4||=?

 

4.Alfred, Beth, and Charles shared a chocolate bar that had a volume of 6 cubic inches. Alfred had 2 cubic inches of chocolate, Beth had 1 of the remaining chocolate, and Charles had the rest.What is the ratio of Alfred’s share to Beth’s share to Charles’s share?

 

5.At a certain travel agency, the number of male customers in May of a certain year increased10% from the previous month, and the number of female customers in that same May decreased8% from the previous month. The number of total customers in May increased by 48 from theprevious month and was 2,148. How many male customers were there that May?

  

Problem Set 6-10

 

6.At an electronics store, a particular TV is purchased from a warehouse and is marked up 40%.The store then decides to reduce the price of the TV by $900, leaving the store with 10% profit.

What is the original price of the TV from the warehouse?

A.$1,000

B.$2,000

C.$3,000

D.$4,000

E.$5,000

 

 

7. A class on a field trip to a park finds identical benches. If 5 students are seated on each bench,there are 3 students standing. However, when they try to seat 6 students on each bench, there is one empty bench and one bench with 4 students on it. How many students are there?

A.57

B.58

C.59

D.60

E.61

 

8.It takes Alfred 24 days to do a certain job by himself. It takes Ben 15 days to complete the same job. If Alfred works on the job for 16 days, and Ben takes over, how many days does it takeBen to finish the job?

A.5

B.6

C.7

D.8

E.9

 

9.There are three consecutive odd integers with the property that the sum of the largest and the smallest numbers is 38 less than 4 times the remaining number. What is the middle number?

A.17

B.19

C.21

D.23

E.25

 

10.What is the least common denominator of the fractions 1/43/10, and 2/21

A.70

B.140

C.210

D.420

E.700 

 

Lesson Complete!

Awesome, awesome to see such effort.

Next, you'll reflect briefly on your past exam. 

If you want to take a break and come back to finish the lesson that's ok!  

 

Test Correction

Self-Correction

Closely examining and analyzing each of your exam performances is one of the simplest, yet most effective ways to improve your ACT score.

On the math section, it’s especially important that you identify the type of errors you are making. How many are silly errors, made in haste?

How many are mis-interpretations of word problems?

How many are simply due to a lack of knowledge of a particular rule?

Throughout this course, you will be asked, once you receive your test results, to review your wrong answers and understand your mistakes. To make this self-correction stick, we ask you here to type an explanation of your math errors for at least 5 problems.

If you got less than 5 wrong on the entire math exam, then woohoo!, simply analyze those few that you got wrong. 

Following your reflection, we will supply you with a full set of answer explanations, so you can see if your corrections were correct! 

Reflect on at least (5) mistakes you made on the math portion of your most recent ACT exam. List the number of the question and then examine the cause of your mistake and how to find the proper solution.

Answer Explanations

 

Lesson Complete!!

Awesome to see such self-reflection.

You're almost done. One pair of brain-teasers and Week 1 is complete!

 

Brain Teasers

Intro

Each Math lesson will close out with a fun challenge that asks you to put what you just learned into practical application. Over the course of the intensive course, your teacher will be keeping track of which students have solved the most teasers. The most successful teaser master will receive a reward at the end of the eight weeks. 

Brain Teaser 1: San Ramon Landscape Architect

Try to answer each part of the question in the space below. 

A. Does it meet the city code?

B. Why or why not? 

Brain Teaser 2: Boomerang

Try to answer each part of the question in the space below. Remember, you can always come back to the question if you can't solve it immediately. 

1.    How much profit will you make each month if 1,000
people buy your boomerangs? Remember, profit is the amount of money you make after you consider costs of purchasing the product.
2.    (a)  How much will your customers save if you offer 25% off?
(b)    How much will they save if you offer buy one, get one free?
3.    (a) How much profit will you make if you offer 25% off of your product and 1,000 people buy your product?
(b)    How much profit will you make if you offer “buy one, get one free” and 1,000 people purchase your product? Remember, if 1,000 people buy your product and they receive one free, they each leave with two products.
4.    Offering 50% off will increase the amount of people who purchase your product by 30%. Is it a wise business decision to do this? Will you profit from this sale tactic?

Week 1 Complete

Really inspiring to see you work through the entirety of the first week!

Whenever you're ready, we will move on to Part 2 of Pre-Algebra fun. Keep up the energy and focus!