GMAT Unit Conversion and Scientific Notation Guide
Matt Kirisits
Website: www.thegmattutor.com
Email: [email protected]
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Table of Contents
1. Unit Conversion ........................................................................................................................................ 3
2. Scientific Notation .................................................................................................................................. 11
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1. Unit Conversion
The key to solving unit conversion problems is to set them up properly. The general strategy is to
cancel units using conversion factors. A conversion factor is the ratio that converts one unit to
another, such as
1 minute = 60 seconds
1 mile = 1.6 kilometers
When using these factors to do unit conversion, write them as ratios. The two conversion factors
above can be written as:
Use conversion factors to set up a multiplication problem in which the unit that you are trying
to cancel is on the opposite side of the ratio. For example, if you begin with kilometers in the
numerator, then put kilometers in the denominator of the conversion factor.
Example 1. A car travels for 46 kilometers. How far did the car travel, in miles? (Note: 1 mile =
1.6 kilometers)
(A) 26.5
(B) 28.75
(C) 32.4
(D) 73.6
(E) 75.25
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Solution:
46 kilometers ×
Set up the calculation
46 kilometers ×
Cancel out kilometers
This problem begins with kilometers in the numerator; thus, put kilometers in the denominator in
the conversion factor. This will make kilometers cancel out, leaving you with miles. Now simply
do the calculation – divide 46 by 1.6.
46 ÷ 1.6 = 28.75 miles
Answer: B
The method of setting up a multiplication problem using conversion factors can be used on all
unit conversion problems. When you set up unit conversion problem in this way, it will prevent
you from multiplying when you should have divided, or vice versa. On the GMAT, it is likely
that the wrong answer choices will test this common mistake. In the above problem, if you
multiplied 46 by 1.6, the result would be 73.6 (answer choice D).
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Unit conversion problems will often involve converting units of time, such as converting hours
to seconds or vice versa.
Example 2. How many seconds are in two hours?
(A) 3,600
(B) 6,000
(C) 7,200
(D) 8,000
(E) 14,400
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Solution:
2 hours ×
×
Set up the calculation
2 hours ×
×
Cancel out hours, then cancel out minutes
The solution is set up to cancel out hours and minutes, leaving you with seconds. After
cancelling the units, do the calculation.
2 × 60 × 60 = 2 * 3,600 = 7,200 seconds
Answer: C
Remember: the unit that you are trying to cancel is placed on the opposite side of the conversion
factor. The example above started with hours in the numerator, so hours was placed in the
denominator of the first conversion factor.
Since converting between hours and seconds is so common on the GMAT, it is useful to
memorize the conversion factor that will allow you to do this calculation in one step:
1 hour = 3,600 seconds
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In the preceding examples, we began with an amount: 46 kilometers in the first example, and 2
hours in the second example. In many cases, we will actually begin with a ratio.
Example 3. A car is travelling at 45 miles per hour. What is its speed in meters per second? (1
mile = 1.6 kilometer, 1 kilometer = 1,000 meters)
(A) 20
(B) 25
(C) 30
(D) 36
(E) 72
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Solution:
You can do this problem in two steps. First, convert miles/hour to meters/hour:
×
×
×
×
= 45 × 1.6 × 1,000 = 72,000 meters/hour
Then, convert meters/hour to meters/second:
×
×
×
×
=
=
= 20 meters/second
Note that this entire conversion could have been set up in one calculation:
×
×
×
×
20 meters/s
Answer: A
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=
=
=
Some problems will be more difficult to set up. Remember to set up your conversion factors to
change the original units into the desired units.
Example 4. The front wheels of a train pass a certain mark 60 seconds before the rear wheels of
the train. If the train is 3,000 feet long, approximately how fast is the train going in miles per
hour? (1 mile = 5,280 feet)
(A) 11
(B) 22
(C) 28
(D) 34
(E) 68
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Solution:
This is a more complicated example. The distance between the front and back of the train is
3,000 feet, so that is the distance travelled is 60 seconds. Start with units of feet/second and
convert to units of miles/hour. Note that the question asks for the approximate rate, so you are
able to use rounding when doing the calculation.
×
×
×
Set up the problem
×
×
×
Cancel out units
=
Write out the calculation
=
Cancel out 60 from top and bottom
=
=
≈
=
= 36
Estimate the solution
The closest answer choice close is 34. The actual calculation of 180,000 ÷ 5,280 would result in
34.1.
Answer: D
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2. Scientific Notation and Unit Conversion
Problems that combine scientific notation problems with unit conversion are very common on
the GMAT. The convention of scientific notation is to write a number as a decimal value with
just one digit to the left of the decimal, multiplied by a power of 10. For example, these numbers
are written in scientific notation:
1.2 × 103
3.26 × 102
8.764 × 105
These numbers are not written in scientific notation:
0.294 × 102
0.0426 × 104
34.86 × 105
However, the second set of numbers above can be converted to scientific notation by moving the
decimal point and changing the power of 10. The rule you must memorize is:
If you are moving the decimal point to the right, decrease the value of the exponent; if you
are moving the decimal point to the left, increase the value of the exponent.
0.294 × 102 = 2.94 × 101
0.0426 × 104= 4.26 × 102
34.86 = 3.486 × 106
Decimal point moves right 1 space, exponent decreases by 1
Decimal point moves right 2 spaces, exponent decreases by 2
Decimal point moves left 1 space, exponent increases by 1
Example 5. The distance between two planets is 8.04 × 1011 miles. What is the distance between
the two planets in kilometers? (Note: 1 mile = 1.6 kilometers)
(A) 1.29 × 1010
(B) 5.03 × 1010
(C) 5.03 × 1011
(D) 1.64 × 1011
(E) 1.29 × 1012
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Solution:
8.04 × 1011 miles ×
Set up the problem
8.04 × 1011 miles ×
Cancel out miles
= 8.04 × 1.6 × 1011
Write out the calculation
= 12.9 × 1011
Multiply out 8.04 × 1.6
= 1.29 × 1012
Move the decimal point one place to the left
and increase the exponent by one
Answer: E
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A slightly more complicated example would require you to divide two numbers that are both
written in scientific notation.
Example 6. The population of a country is 4.08 × 107, and its gross domestic product is 2.02 ×
1011 dollars. What is the country’s approximate per-capita GDP in dollars per person?
(A) 4,950
(B) 20,200
(C) 49,500
(D) 102,500
(E) 202,000
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Solution:
To solve this problem, you should divide gross domestic product by population.
Set up the problem
×
Separate the decimal calculation and the exponent calculation
0.495 × 104
Divide out each component
4,950
Move the decimal 4 places to the right
Answer: A
On this problem, the calculation of 2.02 ÷ 4.08 is time-consuming. Instead of writing out this
calculation, you can estimate the value as or 0.5. You could then simply move the decimal 4
places to the right to get 5,000, which is approximately equal to the correct answer A. Note that
the question asks for the “approximate” per-capita GDP, so it’s fine to use rounding on the
calculation.
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In the example above, the answer choices were written out as integers. If this is the case, you
simply have to write out the full number at the end.
Example 7. At the close of business on a particular day, a bank holds 102 million Swedish
kronor. The closing exchange rate on that day was 1 krona to 0.15 U.S. dollars. What was the
value of the bank’s holdings in U.S. dollars?
(A) 11,150,000
(B) 15,300,000
(C) 68,000,000
(D) 153,000,000
(E) 680,000,000
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Solution:
Note that 1 million is equal to 106.
102 × 106 kronor ×
102 × 106 kronor ×
= 102 × 0.15 × 106 = 15.3 × 106 dollars =
15,300,000 U.S. dollars
Answer: B
In this problem, the answer choices are written out as integers. To remove the scientific notation,
simply move the decimal place to the right by the power of 10. In this case, move the decimal
point six places.
Alternatively, this problem could have been solved by using integer values from the beginning:
102,000,000 kronor ×
= 102,000,000 × 0.15 = 15,300,000 U.S. dollars
The above calculation is approximately 15% of 100 million, which is 15 million. This number is
close to the correct answer choice B.
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