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Going the ‘Distance’: Averages and Rates Problems – Part 2

By posted April 30, 2011

Now that we’re on our way to better scores with our solid grasp of distance question basics, let’s look at two more challenging GMAT questions.

QUESTION #3: Tracey ran to the top of a steep hill at an average pace of 6 miles per hour. She took the exact same trail back down. To her relief, the descent was much faster; her average speed rose to 14 miles per hour. If the entire run took Tracey exactly one hour to complete and she did not make any stops, how many miles is the trail one way?

ANSWER: For the way up the hill, we know that D = 6mph x T.

For the way down the hill, we know that D = 14mph x T.  Since we went know that the distance up the hill was the same as the distance down the hill, we can pick a number for D. Let’s choose “84″ since it is a multiple of both 6 and 14.  If 84 = 6mph x T, then we know that T = 14 hours. If 84 = 14mph x T, then we know that T  = 6 hours.

Now we can use another formula, the Average Rate formula, to find the average speed for the WHOLE trip. Average Rate = Total Distance / Total Time

Using our Picked Number of 84, we know that the Total Distance traveled would be 168 miles. The Total Time is 14 hours + 6 hours = 20 hours.  So the Average Rate = 168 miles / 20 hours = 8.4 mph.

It doesn’t matter that Tracey didn’t “really” go 168 miles, or that we know she didn’t “really” go 20 hours. We Picked a Number just so that we could find the ratio of the Total Distance to the Total Time in order to calculate the Average Rate of the ENTIRE journey.

Now that we have found the Average Rate for the whole trip, we can plug it in to the “DIRT” formula to find the ACTUAL distance for the entire journey.

D = R x T

D = 8.4mph x 1 hour

We know that T = 1 hour because the problem told us so. Therefore, the actual distance for the entire trip was 8.4 miles. The problem asks how many miles the trail was one way. 8.4 / 2 = 4.2. The answer to the question is 4.2 miles.

You could also solve this problem in other ways, including using a system of equations and substitution, but it’s nice to know that you can pick a number for the Distance traveled and use it to find the Average Rate for the whole journey! Be on the lookout for those trips where the distance there and back is the same!

We’ve joked about it, but let’s actually try one of those “train”  questions:

QUESTION #4: A train runs over a straight route from town A to town D. It is scheduled to depart town A at 7am and arrive at town D at 2pm, with 10 minute stops in towns B and C. The train’s top speed is 60mph. The entire length of the route is 320 miles. Will the train arrive on time?

(1) The train experiences a 30 minute delay in town B in addition to its scheduled stop.

(2) The train travels at its top speed for exactly 75% of the trip.

ANSWER: This is a yes/no data sufficiency. In order to answer yes or no, we need to know the time it takes the train to make its journey. From the question, we can see that the train is supposed to take 7 hours to go from Town A to Town D with two 10-minute stops. Thus, the total travel-time of the train is 7 hours – 20 minutes = 6 hours, 40 minutes, or 6.67 hours. A train that takes the full 6.67 hours to travel 320 miles would need to travel at a speed of approx 48mph or faster to make its timetable. Remember that this is what the train is supposed to do. Let’s see how each statement affects the time-table.

With a 30 minute delay, the train’s travel-time is now 6 hours, 10 minutes, or 6.167 hours. With that delay, the train needs to travel at approx 51mph to make its timetable. However, what is missing from this statement is proof that the train actually did increase its speed to make its timetable. Just because it was possible the train arrived on time, doesn’t mean it did. Statement (1) is insufficient.

Let’s look at Statement (2). 75% of the trip is 240 miles out of the total 320 miles. This means the train has to travel the remaining 80 miles in a little less than 3 hours, an average speed of approximately 25 miles an hour in order to make the timetable. This is clearly within the train’s limits, but we do not know anything about the train’s speed for these remaining 80 miles. Statement (2) is insufficient.

Combining the two statements, we know that the train only has 2 hours and 10 minutes to travel the remaining 80 miles. This requires traveling almost 40 mph. This is possible, but we have no certainty that the train accomplished this. The answer, therefore, is (E).

For more practice with Distance and Rate questions, you can schedule a group study game in the GMAT Lobby on Grockit. Use Grockit’s message system to invite other students with high scores and Grockit GMAT tutors to RSVP. Studying together is more fun than studying alone.

 

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Beat the Coordinate Geometry Blues! – Part 1

By posted April 28, 2011

Needing a solid coordinate geometry refresher? The best thing about studying online is that better scores are right at your fingertips. Review these fundamentals and you’ll soon be flying through the GMAT Geometry test questions!

There are two main equations for straight lines. One form looks like:
For an equation that looks like this the slope is and the y intercept is

For example, in the equation 2x + 3y + 6 = 0, the slope is -2/3 and the y-intercept is -2. The second equation is called slope-intercept form and looks like:   Here m is the slope and b is the y-intercept.

Distance Formula =     Use this to find the distance between two points.

Midpoint Formula =

Use this to find the midpoint between two points (notice how you are essentially finding the average of the x-coordinates and the average of the y-coordinates).

Slope = Rise / Run = Change in y / Change in x

Slopes can be positive, negative, zero, or undefined. Positive slopes tilt to the right. Negative slopes lean to the left. A line with a slope of zero is exactly horizontal. The line neither goes up nor goes down as x increases, which is why it has a 0 slope. Vertical lines have undefined slopes. When the two x coordinates are the same, their difference is zero. The slope calculation would leave a 0 in the denominator, which is called “undefined.”

As long as you know any two points on a line, you can find the slope. A line that passes through the origin must have (0,0) as one of its points. Remember that parallel lines have the same slope, and perpendicular lines have negative reciprocal slopes.

Points of intersection can be found by setting two lines equal. What is the point of intersection between the lines y = 2x – 1 and y = -1/2x + 4? Here’s a quick three-step solution:

  1. Put both lines in slope-intercept form. (These example equations are already in slope-intercept form).
  2. Set them equal & solve for x.

2x – 1 = -1/2x + 4

2.5x – 1 = 4

2.5x = 5

x = 2

3.   Plug x back in to either equation to find y.

y = 2x – 1

y = 2(2) – 1

y = 4 – 1

y = 3

The point of intersection will be (2,3). Check out Part 2 for parabolas and circles!

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Going the ‘Distance’: Averages and Rates Problems – Part 1

By posted April 22, 2011

This type of GMAT test question sounds like the beginning of a joke but leaves most students groaning in front of their computers: “A train leaves the station at 4:53am going east at 60mph. A second train….” These word problems are often long, confusingly worded, and just plain boring. The intimidation factor comes from not knowing how to set up the algebra. Let’s look at two must-know formulas that will help boost your test prep confidence.

The first important formula to memorize is: D = R x T. This stands for Distance = Rate x Time. I like to think of it as the “DIRT” formula as fun and easy acronym to remember. It is perfectly acceptable to also think of it as Time = Distance / Rate or as Rate = Distance / Time. 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 an 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 an example question:

QUESTION # 1: I got in my car and drove 40 miles to see my cousin and was going 20 mph. It took me 2 hours to get there. Then, I left my cousin’s and drove another 30 miles to the store but this time went 10mph. It took me 3 hours to arrive at the store. What was my “Average Speed” for the whole trip?

ANSWER: Average Speed = Total Distance / Total Time. I traveled 40 miles + 30 miles so my Total Distance was 70 miles. I drove for 2 hours + 3 hours so my Total Time was 5 hours. 70/5 = 14. My Average Speed for the whole trip was 14 mph.

The Average Speed in this problem is 14 mph, which is different from the “Average of the Speeds.” If we had just averaged the two speeds (10mph and 20mph) we would have gotten 15mph. Think of Average Speed as a weighted average. I spent more time in the problem going 10mph than 20mph, so it makes sense that the Average Speed would be closer to 10mph. Be careful because the “Average of the Speeds” will often be a tempting wrong answer choice!

Let’s try another one that will require us to use both the “Average Rate” formula and the “DIRT” formula.

QUESTION # 2: Marion spent all day on a sightseeing trip in Tuscany. First she boarded the bus which went 15mph through a 30 mile section of the countryside. The bus then stopped for lunch in Florence before continuing on a 3 hour tour of the city’s sights at speed of 10mph. Finally, the bus left the city and drove 40 miles straight back to the hotel. Marion arrived back at her hotel exactly 2 hours after leaving Florence. What was the bus’s average rate for the entire journey?

ANSWER: To find the “Average Rate” of the bus, we know we will need to find the Total Distance and the Total Time, so let’s see how we can use the D = R x T formula to find the missing info.

For the first part of the trip, we know that 30 miles = 15mph x T, so we know that T = 2 hours.  For the middle part of the trip, we know that D = 10mph x 3 hours, so we know that D = 30 miles. For the last part of the trip, we know that 40 miles = R x 2 hours, so we know that R = 20mph.

Now we can find the Total Distance and the Total Time. Total Distance = 30 miles + 30 miles + 40miles = 100 miles. Total Time = 2 hours + 3 hours + 2 hours = 7 hours.

So the Average Rate = 100 miles/ 7 hours = 14.28mph.

Check out Part 2 for more challenging questions!

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Evaluating Expressions – Reviewing the GMAT Basics

By posted April 18, 2011

All GMAT test prep must start with comprehensive content review. Don’t feel guilty if you can’t recall the difference between an “integer” and a “whole number.”  Below is a quick overview of basic number properties and arithmetic operations to jump-start your online studying. Don’t ignore the small stuff; these topics are the building blocks for better scores!

A variable is a symbol representing a numerical quantity. Variables are represented by letters in the alphabet such as x, y, a, b etc. The number that the variable represents is called a value.

A constant is a symbol that represents a definite quantity (such as 3 or -3).

A term is a product (multiplication) made up of either variables or constants with an unspecified number of factors. Terms that have the same factors which differ only in the number in front of them are called similar terms. For example, 5y and 9y are similar terms since they both contain a y, but one has a 5 in front of it, and one has a 9 in front of it.

An algebraic expression is a mathematical statement which combines constants and variables. For example: 75x + 12 is an algebraic expression.  So is 3x + 4.

Because of the distributive property, similar terms can be combined into one term. The new term has the same factors as the similar terms, but its coefficient is the sum (addition) of the coefficients; this is commonly known as combining like terms. Using the example above, 5y + 9y = 14y.  Similarly, 3xy + 2xy = 5xy

PEMDAS is an acronym for the order of operations, which stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. Always start with what is inside the parentheses then address any exponents to simplify an expression. Next, move left to right, doing all division and multiplication. Finally, again moving from left to right, do any addition or subtraction.

Number Types

Integers: All numbers that are multiples of 1 with no fractional or decimal parts.

Whole numbers: Positive integers (includes 0). These are also called the “counting numbers.”

Real numbers: All numbers on the number line.

Imaginary numbers: i is the most common. i = √-1, so i^2=-1 .

Rational numbers: All numbers that can be expressed as a ratio of two integers. Decimals and fractions are rational numbers if they have a terminating or repeating decimal. For example, 1/3 is rational, as it repeats to .33333, but π is not.

A few operations to keep in mind:

Subtracting negative numbers: When you subtract a negative number, you will add the terms. Example: 5 – (-2) = 5 + 2 = 7

Zero: Dividing by zero is undefined. The denominator of a fraction cannot be zero. This concept is commonly tested on the GMAT. 1/0 = undefined.

Recpirocals: The reciprocal of a number is 1 divided by the number. For a fraction, the reciprocal can be determined by flipping the numerator and the denominator. A number times its reciprocal = 1.

Squaring Fractions: If you square a number between 0 and 1, the number gets smaller. For example, (1/2)2 =1/4.

Multiplying & dividing negatives: The product or quotient of two numbers with the same sign is positive, even if both numbers are negative. (-4) x (-2) = 8.

 

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Powers & Roots: Simple Tips for Better Scores

By posted April 8, 2011

Even if you know the basic rules for questions involving powers and roots, (click here for a quick review),  it’s still common to feel some intimidation towards harder-looking questions.  Let’s look at a few tips that will help you focus your test prep when dealing with more challenging test questions.

1. For variable questions, always pick numbers!

Don’t assume you know what the outcome will be, even if you’re the Stephen Hawking of data sufficiency. Pick numbers every time; actually seeing how the numbers work out when square, cubed, etc. will give you more confidence. Let’s check out a Grockit question:

Is the following statement true: [pmath]y^2 > y[/pmath]?

(1) [pmath]y^2 > 4[/pmath]

(2) [pmath]y > -2[/pmath]

Because you know your exponent rules, you know that whether [pmath]y^2 > y[/pmath] is true is dependent on what kind of number y is. Let’s use Tip 1, and quickly plug in:

If y = 1, then [pmath]y^2 > y = 1 > 1[/pmath]. Not true.

If y = [pmath]1/2[/pmath], then [pmath]y^2 > y = 1/4 > 1/2[/pmath]. Not true.

If y = 0, then [pmath]y^2 > y = 0 > 0[/pmath]. Not true.

If y = [pmath]-1/2[/pmath], then [pmath]y^2 > y = 1/4 > -1/2[/pmath]. True.

If y = -1, then [pmath]y^2 > y = 1 > -1[/pmath]. True.

We can see that when y is negative, the answer will be YES. When y is positive, the answer will be NO.

2. Harder questions combine concepts.

The rules don’t change because powers and roots are combined with a secondary concept, so don’t let a combination of ideas throw you. Now let’s go back to the previous question and look at the statements. This question is adding the concept of inequalities to the mixture.

Based on our previous analysis, we know that in order for a statement to be sufficient in this question it must tell us that y is either positive 100% of the time, or negative 100% of the time (the actual outcome to the question doesn’t matter).

(1)  [pmath]y^2 > 4[/pmath]

(2)  [pmath]y > -2[/pmath]

We can solve Statement 1 by taking the square root of both sides. Remember that for inequalities, we’ll have two solutions: y > 2 or y < -2. The secondary concept here is that we must flip the inequality for the negative solution. If you were unsure, you could plug in a few values: 3, -3, etc. and you would see that both satisfy [pmath]y^2 > 4[/pmath]. Since there is no overlap between these two solution sets then y can only be EITHER greater than 2 or less than -2. No matter which solution set y comes from, it will give only one answer, since y > 2 means the statement is not true, and y < – 2 means the statement is true. Therefore, Statement (1) is sufficient.

Now let’s examine Statement (2). y > -2. This solution set includes all numbers greater than -2. This set encompasses negative AND positive numbers, so depending on the value of y, this would make the statement sometimes true and sometimes false. This means (2) is insufficient. The answer is (A).

When you study online, it’s easier to forget about a question once you’ve answered it, but you may find it helpful to review old questions and write down the “secondary” concepts that are commonly tested with Powers & Roots. You’ll find certain concepts are paired together more than others, and you’ll start to see how the concepts are commonly presented. You can check out the full explanation for this question and others in Grockit’s new History section in our redesigned lobby. Happy studying!

Grockit

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Use Strategy to Improve your Score!

By posted March 29, 2011

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 Grockit group game is NOT the same as actually having a solid strategy.

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 Grockit, 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.

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GMAT Quantitative: Quadrilaterals

By posted March 18, 2011

On the GMAT, you’ll have to be familiar with your shapes. Unfortunately, this knowledge goes significantly beyond distinguishing a circle from a triangle, though you’ll still have to know that.

Let’s talk about a popular one: quadrilaterals. “Quadrilateral” is just a fancy, polysyllabic word  for any four-sided polygon. There are a few different types of quadrilaterals that you should be familiar with, but first, I’ll discuss some basic properties that all quadrilaterals have in common.

Universal Properties

  • All interior angles of a quadrilateral add up to 360 degrees.
  • Find the perimeter of any quadrilateral by adding up the four sides.

There are three basic types of quadrilaterals you will find on the test: parallelograms, rectangles, and squares.

Parallelograms

By definition, a parallelogram is any quadrilateral whose opposite sides are both parallel and equal in length. Notice in the diagram that the opposite angles are equal, and the consecutive angles–those  that share a side–are supplementary (they add up to 180 degrees).

Area of a Parallelogram: base times height, or bh. (Simpler than it looks)

2

Rectangles

Insert image: http://www.domesatreview.com/images/satmath/quad-2.png

Rectangles are special parallelograms in which all the angles are right angles. Note that they still retain the characteristic that opposite sides are equal.

Area of a Rectangle: base times height, or bh.

Squares

A square is a special type of rectangle in which all the sides are equal.

Area of a Square: base times height, or side^2

Special Information

Granted, the GMAT will not simply test you on your ability to spot these special types of quadrilaterals. You’ll have to use your knowledge of their properties to perform measurements and calculations. Here are some familiar situations for quadrilateral measurement:

1.      Diagonals

You’ll often have to find the diagonal of a rectangle to yield new information about a shape. The diagonal is just the line that extends from one corner to another, and you can calculate a rectangle’s diagonal by using the Pythagorean Theroem: a^2 + b^2= c^2. All rectangles are made up of two congruent right triangles. In fact, all squares are made up of two special congruent right triangles called 45-45-90 triangles. These special isosceles triangles are so named because they are made up of one 90 degree and two 45 degree angles. To save yourself time, remember this diagram:

With this information, you could find out the sides of a square (and therefore its area and perimeter) with just its diagonal measurement. For example, if the diagonal of a given square is 5, then I can use this equation to yield side n:

n*rad(2) = 5

n=5 / rad(2)

rationalize the denominator…

n=(5 *rad 2 ) / 2

2.      The Square in Circle Problem

You may see a square inscribed in a circle like the one above, and the question may ask you for the area of the square provided that the radius of the circle is 2.5. To solve this problem, remember that the diameter of this circle must be equal to the diagonal of the square–draw a line from two opposite corners and see for yourself. To solve it, just recognize that a radius of 2.5 yields a diameter of 5, which also happens to be the length of our diagonal. Use the steps above to find side n¸and simply square that n.

Those are the quadrilateral basics. Whenever you’re stumped on a word problem with measurements, remember the properties of quads and it may get you out of a jam!

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Area Refresher Part 2

By posted March 16, 2011

Trapezium – The GMAT often tests a student’s ability to calculate the area of a trapezium, a shape that is slightly more difficult than the standard shapes I mentioned earlier, as it sometimes requires calculating the area of two such standard shapes. A trapezium is represented as follows:

The two sides of a trapezium are parallel as shown. However the other two sides can take any shape and length. The general formula for calculating Area of a trapezium is as follows:

Area = ½ x (Sum of two parallel side lengths) x Height of Trapezium.

The trick to remember this formula is that trapezium is really made up of two triangles.

And as discussed earlier, the area of any triangle is ½ x Base x Height. Therefore, the above trapezium formula is sum of the areas of the two triangles.

Circle – A circle represents an enclosed shape where each point on the circumference of the circle is equidistant from the center of the circle. The distance from the center to any point on the circumference is called Radius of the circle.

Area of a circle is given as a function of the radius of the circle.

Where Π is a constant with value = 3.14

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GMAT Practice: A Process for Problem Solving

By posted March 15, 2011

Problem Solving is often overlooked strategy-wise, since it’s easy to place more emphasis on developing a process for the unfamiliar question-type, Data Sufficiency. However, it’s important to make sure you have a methodical step-by-step approach to these more traditional math questions. Even if you know how to do the math, reading too quickly, or jumping into the algebra without really understanding the question can lead you to the incorrect answer.

1.  What’s it really asking? Write down what the question is asking you to find. It may sound obvious, but the GMAT often requires an extra step. You may need to find 1/y, instead of y. Or you may be asked about the “ratio of girls in a class to boys in a class,” but have to solve for the two parts of the ratio first. If you don’t write it down, you will not see the end goal as clearly.

Need one-on-one GMAT help? Write to one of Grockit’s GMAT experts and find out how you can set up customized tutoring on Grockit.

2.  What’s the important info? Pull out any key numbers, variables, or phrases from the question and write them down on your scratch pad. This is the step most students skip. Don’t just scan the screen and start solving. Forcing yourself to slow down and process each piece of information will give your brain time to sort through it. This is especially important for word problems which can contain unnecessary extra information. This may lead you to find a faster way to solve!

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Area Refresher Part 1

By posted March 14, 2011

A sizeable number of GMAT math test questions belong to the Geometry section.  Some of these questions test  a candidate’s ability to understand 2-Dimensional Geometry by asking the candidate to calculate the area, perimeter or circumference of a geometrical shape.

The following geometrical shapes are most common – Triangles, Quadrilaterals, Rectangles, Rhombuses, Squares, Circles and Trapeziums.

Triangles – A triangle represents an enclosed shape made by joining three straight lines. The area of a triangle can be calculated as follows:

Area = ½*Base Side*Height of the triangle

In this formula, the Base Side can be any side of the triangle. However, depending on the base side chosen, height of the triangle needs to be ascertained. Height of the triangle is the shortest perpendicular distance from the Base side to the height of the Apex of that triangle.  Note that the height of a triangle may need to be calculated outside the triangle, depending on the base side chosen.

Triangles can be of different shapes depending upon the length of each of the three straight lines that create the triangle. For example, if two lines are of equal length then it results in an isosceles triangle. If all three sides are equal, it results in an equilateral triangle. A scalene triangle has three sides that are of unequal lengths.

Quadrilateral – A quadrilateral represents an enclosed shape made by joining four lines. The GMAT questions will ask the test taker to calculate areas for the following types of quadrilaterals, each of which has a specific formula for Area:

Square – A square is a quadrilateral that has four equal sides, and each side is perpendicular to two other sides of the square. The area of the Square is given by:

Rectangle – Rectangles have special properties:  each side connects to two other sides perpendicularly, and the parallel facing sides are equal. The area of a rectangle is given by:

Area = product of two adjacent sides

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