Tag Archives: math

Interpreting Evidence Correctly on the ACT and SAT

In all three types of ACT Science passages, you will need to be able to understand how evidence is presented and be asked to answer questions based on the evidence. These are a bit like Reading questions.

To interpret evidence correctly, you need to focus on the results. Draw logical conclusions based on the presented data, and actively read the information in the accompanying passage. It’s important to know the difference between direct and inverse variation.

To answer Interpreting Evidence questions correctly, ask yourself these 3 questions: – What is the evidence presented? – Whose position is supported by the evidence? – What does the evidence suggest?

Interpreting evidence for Data Representation and Research Summaries ends up requiring more Data Analysis skills, since data is a large component of those passages. For Conflicting Viewpoints, interpreting evidence is a matter of keeping the two theories (two scientists) straight. Watch this video for a strategy on how to do this!

This video describes how to answer “Inference” questions on the SAT and ACT Reading Test. You may wonder what this has to do with ACT Science, but sometimes interpreting evidence is a LOT less scientific than it sounds, and a LOT more like reading comprehension. Remember that you can only make an inference BASED on something directly stated. Incorrect answer choices are frequently “out of scope.”

 

3 Ways of Looking at “Profit” Questions on the GMAT

As someone who is about to shell out hundreds of dollars in MBA application fees, you know that money makes the GMAT-world go round. Profit is an essential concept for any aspiring MBA admissions applicant. The GMAT tests this concept in both Problem Solving and Data Sufficiency questions in three main ways. Let’s examine the need-to-know formulas with three GMAT practice questions.

1. A firm increases its revenues by 10% between 2008 and 2009. The firm’s costs increase by 8% during this same time. What is the firm’s percent increase in profits over this period, if profits are defined as revenues minus costs?

(1) The firm’s initial profit is $200,000.

(2) The firm’s initial revenues are 1.5 times its initial costs.

In this question from Grockit, we can start with our most basic Profit formula:

Profit = Revenue – Cost

Using Statement (1), we can say that 200,000 = R – C.
(1.1)r – (1.08)c = 200,000(1 + x), where x equals the amount of the increase. We still do not know R and C so we can’t find x. Insufficient.

Using Statement (2), 1.5c – c = p and (1.1)(1.5)c – (1.08)c = (1 + x)P. Here we can simplify.

.5c = p

.57c = (1 + x)p
Without continuing to solve, we can see that we can solve for x using substitution. .57c = (1 + x)(.5c), and dividing both sides by c will cancel out that variable and allow us to isolate x. Statement 2 is sufficient. Now to a more challenging question!

2. A store purchased 20 coats that each cost an equal amount and then sold each of the 20 coats at an equal price. What was the stores gross profit on the 20 coats?

(1) If the selling price per coat had been twice as much, the store’s gross profit on the 20 coats would have been $2400.

(2) If the selling price per coat had been $2 more, the store’s gross profit on the 20 coats would have been $440.

This GMAT Prep question asks about gross profit.

Gross Profit = Selling price – Cost

For the value Data Sufficiency question, we need to know the price of each coat and the selling price of each coat. From the given information, we can use our known formula to set us the equation: P = 20 (s – c). So either we’ll need a value for s and a value for c, or we’ll need the value of (s – c).

Statement (1) tells us that $2400 = (20(2s – c)) or 2400 = 40s – 20c. We can divide both sides by 20 and simplify it to: 120 = 2s – c. We still don’t know s and c. Insufficient.

Statement (2) tells us that 440 = 20(s + 2 – c). Let’s simplify: 440 = 20s + 40 – 20c. 400 = 20s – 20c. 400 = 20 (s – c). 20 = s – c. Sufficient. Even though we didn’t solve for s and c separately, we were able to find the value of (s – c). Sometimes DS will surprise you!

3. If the cost price of 20 articles is equal to the selling price of 25 articles, what is the % profit or loss made by the merchant?

A. 25% loss
B. 25% profit
C. 20% loss
D. 20% profit
E. 5% profit

Profit/Loss % = (Sales Price – Cost Price) / Cost Price x 100

The question asks about % profit or loss. It tells us that 20c = 25s, or 4c = 5s. So the ratio of the sales price to the cost price is 4/5.

Let’s simplify our Profit/Loss % formula by dividing each term by the cost price:

Profit/Loss % = (S/C – C/C) x 100

P/L% = (S/C – 1) x 100
We know that S/C = 4/5 for this problem. So we can plug in and solve:

P/L% = (4/5 – 1) x 100

P/L% = (-1/5) x 100

P/L% = -20%. The answer is a 20% loss.

Using Strategy on the GMAT to Improve your Score

When a GMAT student asks me, “What can I do to get better scores?” usually the first thing I ask is, “What is your current strategy?” Most of the time, I get a pretty vague response. Reading about strategy is the OG, on the BTG forum, or in a GMAT book is NOT the same as actually having a solid strategy. The word “strategy” may sound fuzzy, but all it means is a simple step-by-step approach for each unique question type.

Not only do you have to choose a strategy that works for you, but you have to implement it every time, practicing enough so that is becomes second-hand. Ballet dancers practice a pirouette millions of times, so that when they perform onstage they don’t have to think about it. You want to do the same thing for GMAT.

Before you sit down to take your next diagnostic on GMATPrep, quickly review this strategy cheat sheet (or make one of your own). These methods may not necessarily work for you, but you’ll only learn what does through trial and error. For more in-depth discussion on each of these strategies, search my other posts.

Verbal

Reading Comprehension –

1. Break down the passage. 2. Rephrase the question. 3. Predict an answer. 4. Eliminate.

Critical Reasoning –

1. Identify the Conclusion, Evidence & Assumptions. 2. Rephrase the question. 3. Predict and answer.

Sentence Correction –

1. Spot the primary error. 2. Eliminate answer choices that do not fix. 3. Look for secondary errors and eliminate.

Quant

Problem Solving –

1. Write down the given information. 2. Scan the answer choices. 3. Look for ways to pick numbers or plug in. 4. Recall relevant formulas. 5. Solve.

Data Sufficiency –

1. Identify the type of DS. 2. Determine what is needed for sufficiency. 3. Evaluate statements independently. 4. Combine if needed.

700+ GMAT: Rock Set Theory

Venn diagrams and matrices getting you down? No clue what “elements” are? Sets on the GMAT have a reputation for being tough, but that’s just because most students are less familiar with them. This GMAT board will fill you in on the basics!

The “Intersection” is an upside-down U symbol, and is the OVERLAP of the sets. That is, the intersection contains all the elements that are in BOTH sets. Notice the Venn diagram is used to show the Intersection.

It makes sense that the symbol for “Union” would be a “U” shape. The Union is always the total combined elements. If an element is in EITHER of the sets, then it’s in the Union.

Sometimes Sets questions will be combined with other concepts, such as percentages. They often will not require fancy Venn diagrams or the ability to use a matrix to solve. Watch this Grockit video to see an example of this. You probably didn’t even know this could be considered a “sets” question! 🙂

Like a Venn diagram, a Sets Table (or matrix) is a great way to systematically organize a lot of information, especially for a Sets word problem. Read through this blog on how to set one up! Notice how the table is set up 3 x 3.

 

Two Types of GRE “Averages”: Mean and Rates

The word “average” on the GRE can refer to two concepts: arithmetic mean, and the average speed (or average rate) formula. It’s important not to confuse the two on the Test Day, as they require different formulas to solve.

Mean is the mathematical average. This is defined as the sum of the terms divided by the number of terms. Mean = Sum / # of terms. For a list of consecutive integers or evenly spaced numbers, the mean is equal to the median, or the middle number. For example, the “average” of 3, 5, and 9 is 5.67.

Average Speed or Average Rate is often found in complex word problems. This type of question is one many students are less familiar with so you may not have seen it before. Let’s review two important equations to remember and look at how this concept appears on the GRE.

The first formula to memorize is: D = R x T. This stands for Distance = Rate x Time (referred to as the “DIRT” formula). It is perfectly acceptable to also think of it as Time = Distance / Rate or as Rate = Distance / Time as well. Usually the “Rate” is speed but it could be anything “per” anything. In a word problem, if you see the word “per” you know this is a question involving rates.

The second formula is: Average Rate = Total Distance / Total Time. This is its own special concept and you will notice that it is NOT a simple Average of the Speeds (which would be something like the Sum of the Speeds / the Number of Different Speeds or what we know as the Arithmetic Mean). Average Rate is a completely different concept, so do not let the common word “average” confuse you. Let’s look at a sample question from Grockit’s GRE question bank:

Question 1: The average (arithmetic mean) of four numbers is 30, after one of the numbers is removed, the average of the remaining three numbers is 10. What number was removed?

We know that the four original numbers sum to 30*4 = 120. The new equation becomes:

4*30 – x/3 = 10
120 – x/3 = 10
120 – x = 30 (add an x to each side and subtract a 30)
90 = x

Ratios and Proportions on the GMAT

A ratio expresses the relationship between two or more things. A proportion is a relationship that is formed by setting two ratios equal. Learn how to solve proportion problems using equivalent ratios on the GMAT….like a rockstar on this Learnboard!

Once you’ve reviewed the board, try this Data Sufficiency problem on your own:

For each month, the number of accounts, a, that a certain salesman has contracted that month is directly proportional to his efficiency score, e, which is directly proportional to his commission rate, c. What is a if c = 3.0?

(1) Whenever c = 4.0, e = 0.3

(2) Whenever c = 6.0, a = 80

Explanation:

It will be helpful to first note that because a is directly proportional to e, which is in turn directly proportional to c, a is then directly proportional to c. To say that a is directly proportional to c is just to say that there is a constant k such that ck = a, or, perhaps more simply, that there is a fixed ratio between a and c. A statement, or set of statements, will be sufficient if and only if it determines that ratio.

Statement (1): From this, the proportional relationship between e and c can be determined. However, a is directly proportional to e, and nothing is said about that relationship; therefore, the value of a when c = 3.0 cannot be found; NOT sufficient.

Statement (2): This gives you the ratio you want. You don’t need to actually calculate the value of a if c = 3.0. You just need to know that it’s possible. Don’t believe me? Because a is directly proportional to c: a/c = 80/6.0. Since the question asks for the value of a when c = 3.0, divide the numerator and denominator each be 2. a = 40. Or, if you’re determined to cross multiply: substitute the given value for c: a/3.0 = 80/6.0. By cross multiplication, 6a = 240. Therefore, a = 40; SUFFICIENT. The credited response is B.

Tricky Triangles: the GMAT’s Favorite Shape!

A triangle is a three-sided shape whose three inner angles must sum to 180°. The largest angle will always be across from the longest side. Triangles are the most commonly-tested geometry topic on the GMAT!

Remember the sum of all the interior angles in a triangle will sum to 180 degrees, so you can always solve for the third angle if you know the other two.

If you’re told two triangles are similar, the corresponding angles are congruent, or equal. You can set up various proportions to the corresponding sides as well.

More essential info:essential info: the side of any triangle must be BETWEEN the sum and the difference of the other two sides.

Check out some practice problems to refresh your triangle properties on this Learnboard!

Breaking the GMAT Word Problem Barrier!

“Expect problems and eat them for breakfast.”

– Alfred A. Montapert

Word problems on the GMAT get an unfair reputation for being especially challenging. Once you’re able to effortlessly translate the key phrases into algebra, you’ll be able to handle any word problem with aplomb!

Here’s my favorite resources from around the web all in one convenient Learnboard to help you overcome your fear of word problems: GMAT – Word Problems. And, no worries if Word Problems are the bane of your existence, we’ll start with the very basics!

Learnist: Simplifying Algebraic Expressions on the GMAT

Algebra is fundamental to GMAT Quant. A great way to get started on your GMAT prep is to refresh your skills in simplifying algebraic expressions!

PEMDAS is an acronym for the order of operations, which are the basic rules which govern the simplification of algebra. Notice how division/subtraction is always done in order from left to right.

Addition and multiplication are both “commutative” which means it doesn’t matter the order in which the operation is performed. This means that A + B = B + A, and A x B = B x A.

The Associative Property for addition and multiplication means that the numbers can be re-grouped in parentheses without a different outcome. For example, 2 + (3 + 7) = (2 + 3) + 7. Like the Commutative law, this is ONLY true for addition and multiplication.

The Distributive law allows us to “distribute” a factor among terms being added or subtracted. That is, a(b + c) = ab + ac. This law, along with the commutative and associative laws, will become second-nature to you the more you practice!

Remember this rule: you can ONLY cancel factors. Try to simplify the numerator and the denominator as much as possible if you’re looking for things to cancel.

Notice that algebraic expressions can be made more complicated with exponents, including negative exponents. Remember your exponent rules! When you have the same base in the numerator and the denominator, you can subtract the exponents.

Watch some video walk-throughs of some GMAT algebra problems involving order of operations and algebraic expressions on the GMAT – Simplifying Algebraic Expressions learnboard.

Contrast Words in GMAT Sentence Correction

Contrast keywords are fairly common on GMAT Sentence Correction questions. Some of the most common words you’ll see include:

  • but
  • however
  • on the other hand
  • unlike
  • in contrast to
  • different from
  • whereas
  • while
  • yet
  • as opposed to

Let’s look at an example question now

However a minimal role in causing air pollution, trash pollutes both land and sea, creates problems in wildlife habitats, and harms endangered species

A) However
B) With
C) Being
D) Therefore accounting for
E ) Despite playing

—

Here we have the phrases “minimal role” and “trash pollutes..creates problems…harms…” which tells us we’ll need a Contrast word to open the sentence. (B), (C), and (D) do not provide a contrast, so we can quickly eliminate them.

Between (A) and (E), (E) is the better choice since “However a minimal role…” is an awkward construction. A better way to use “however” here would be: However minimal its role….