Question 1

Use a calculator to find the sine of 46.3°

Accepted Answer

The sine of 46.3° is0.723
Solution (steps taken are dependent on type of scientific calculator used)
Enter 46.3 into the calculator and press the SIN key.
The answer 0.723 will be displayed.

Question 2

When formulating answers to market-sizing questions, what's one of the best strategies to keep in mind?

Accepted Answer

Estimating and rounding numbers is a key skill in getting through questions as quickly as possible.
Key Takeaway: Before formulating your answer, take a moment to choose your numbers carefully - be sure to pick rounded numbers that work easily together, making your arithmetic much less of a headache.

Question 3

Given a triangle with sides a = 20 feet, b = 40 feet, and c = 56, what is the area in square feet?

Accepted Answer

The area of the triangle is 281.68 ft². Solution Use the formula for the area of a triangle with sides a, b, and c: A = [s(s-a)(s-b)(s-c)]½ where s = (a + b + c)/2 Where a = 20 ft b = 40 ft c = 56 ft s = (20 ft + 40 ft + 56 ft)/2 = 58 ft A = [58 (58-20)(58 - 40)(58 - 56)]^½= 281.68 ft²

Question 4

Shane's Softball Production Company can produce 12 softballs in an hour. How many softballs can they produce in 14 hours?

Accepted Answer

168 softballs. Although this is not a difficult problem, and could be calculated mentally, perhaps by multiplying 14 by 10 (140), then adding 2 more 12s to the answer.
However, there is a trick that allows you to multiply numbers close to ten together within seconds.
First, consider the problem: 14 x 12
Using the 2 in 12, add it to 14: 16 (these are the first two digits of the answer).
Using the 2 in 12, multiply it by the 4 in 14: 8 (this is the last digit of the answer).
Thus, the answer to 14 x 12 is 168.
It works for larger numbers in the teens, but those involve "carrying".
Consider the problem: 17 x 14.
Using the 4 in 14, add it to 17: 21
Using the 4 in 14, multiply it by the 7 in 17 (28). The 8 is the last digit of the answer. Carry the 2 over, and add it to the 21 from the previous step to get 23. The answer to 17 x 14 is 238.
Key Takeaway: Sometimes you will find it necessary to multiply numbers in the teens together. With practice, this strategy will allow you to perform that calculation mentally, both quickly and accurately.
Again, this is a trick that requires lots of practice that works for some, and doesn't work for others.

Question 5

The given numbers are daily groundwater elevation readings from a monitoring well taken during a month experiencing snowmelt and extended rainfall events.7.6, 7.7, 4.3, 5.9, 5.0, 10.5, 7.7, 9.5, 12.0, 12.6, 6.5, 8.3, 8.2, 13.2, 12.6, 13.6, 14.1, 13.5, 11.5, 12.0, 10.4, 10.8, 13.1, 12.5, 10.4, 13.0, 14.1, 15.1What is the sample variance of these groundwater elevation readings?

Accepted Answer

The sample variance of the groundwater elevation readings is 9.089. Solution Use the sample variance (s²) equation s² = ?(sample value - sample mean)2/(number of samples-1) Where Number of samples = 28 Sample mean = ?(sample values)/number of samples = 295.7/28 = 10.56 ?(sample value - sample mean)² = (7.6-10.56)²+(7.7-10.56)² + …… = 245.4 s² = 245.4/27 = 9.089

Question 6

A small firm allocates 28% of its gross profits into an account for insurance. Last year, the firm's gross profit was $400,000. What amount was deposited into the insurance account?

Accepted Answer

To calculate this mentally, use this idea: x% of y = y% of x. Therefore, 28% of 400,000 is the same as 400,000% of 28. Multiply 4 x 28 to get 112, then add the zeroes back at the end ($112,000). You know that this is a reasonable answer because 28% is a little over ¼, and ¼ of 400,000 is 100,000.
Key Takeaway: When you are calculating percentages of round numbers, it is helpful to know that x% of y is the same as y% of x.
Using this, you are often doubling, tripling, or multiplying by a single number, then adding the zeroes back at the end.
This takes some practice, but it works well for some people.

Question 7

Convert 75°13'22" to decimal degrees.

Accepted Answer

75°13'22" is equal to 75.223° in decimal degrees.
Solution
One minute is defined as 1/60 of a degree (1° = 60 minutes)
One second is defined as 1/60 of a minute (one minute = 60 seconds)
Therefore, 1° = (60 minutes per degree) (60 seconds per minute) = 3600 seconds
1. Convert the seconds to degrees
(22 seconds)(1° per 3600 seconds) = 0.006°
2. Convert minutes to degrees
(13 minutes) (1° per 60 minutes) = 0.217°
3. Add together to obtain decimal degrees
75.00 + 0.006 = 0.217 = 75.223°

Question 8

Seven/Twelve has $4 million in sales over the past week. What were the approximate sales per day?

Accepted Answer

$571,000. $4 million divided by 7 days is an average of $571,429 per day.
Key Takeaway: Dividing by seven is an important skill because of the week unit is frequently used. If you remembered the repeating decimal we gave you earlier, all you have to do is move it over in this question.
1/7 = .142857
2/7 = .285714
3/7 = .428571
4/7 = .571428
5/7 = .714285
6/7 = .857142

Question 9

If the speed of a car is 200 km per hour, how may minutes does it take to travel 60 km?

Accepted Answer

18 minutes. The quickest way to do this problem is to divde 60 by 200.
You can quickly reduce that to 3/10. Then multiply 3/10 by 60 to get 18 minutes.
Key Takeaway: Again, we can't emphasize the value of being really comfortable with the time units. 60 breaks down nicely with fractions as it is divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30.

Question 10

What is 1/7?

Accepted Answer

Key Takeaway: If you memorize the sequence 1-4-2-8-5-7 you can calculate any number divided by 7 quickly.
1/7 = .142857
2/7 = .285714
3/7 = .428571
4/7 = .571428
5/7 = .714285
6/7 = .857142
Editor's Note: In one of my case interviews, I was asked to calculate how many $70 devices I would need to reach $2 million in revenue. Since someone had taught me this trick, I dispensed with estimating, and told the interview the answer was 28,571.428571. I got the job.

Question 11

Last year, Smith's Auto Shop made gross revenue of $460,000. This year, their gross revenue increased significantly - by an amazing 25%. What was the gross revenue this year?

Accepted Answer

To find 25% of a given number, simply halve the number, then halve it again. In this example, half of 460,000 is 230,000, then half of 230,000 is 115,000. Therefore, 25% of $460,000 is $115,000. Add $115,000 to the original $460,000 to find the gross revenue for the year - $575,000.
Key takeaway: It is helpful to know shortcuts for mentally calculating certain percentages. To find 10% of a number, drop the last digit. To find 20% of a number, drop the last digit, and double the resulting number. To find 25% of a number, halve the number twice.

Question 12

Given a triangle with a base of 42.20 feet and a height of 63.10 feet, find the area of the triangle.

Accepted Answer

The area of the triangle is 1,331.41 ft². Solution Use the formula A = 1/2bh Where A = area b = base h = height A = ½ (42.20 ft) (63.10 ft) = 1,331.41 ft²

Question 13

The population of the town of Homeville increased by approximately 0.5% last year. If the population at the beginning of the year was 2,460, what was the approximate population by the end of the year?

Accepted Answer

This problem is asking you to find ½ of 1 percent change of 2,460. To do this mentally, first determine what 1 percent of the original number is, then half that. In this case, 1% of 2,460 is 24.6, and half of that is 12.3. Since you can't have 3/10 of a person, we can approximate the population increase as 12 people. Add that to the initial 2,460 to find that the new population is 2,472.
Key takeaway: There are many types of percent problems you may encounter. It will be helpful to you if you know that you find 10% of a number by moving the decimal one place to the left and 1% of a number by moving the decimal point two places to the left.

Question 14

What is 5/6?

Accepted Answer

5/6 is equal to .8333. The 3s repeat forever.
Key Takeaway: Memorize 1/6 and 5/6 to speed your calculations up (trust me, it's worth it).
1/6 = .166666
5/6 = .833333

Question 15

Which of the following is a Double Angle Identity formula?

Accepted Answer

Sin 2x = 2 sin xcosx is a Double Angle Identity formula.
Double Angle formulas are useful when integrating rational functions of sine and cosine.
Sin (x + y) = sin xcosy - cosx sin y is a Sum and Difference Identity formula
Sin xcosy = ½ [sin(x + y) + sin (x - y) is a Product-Sum Identity formula

Question 16

In a game, one card is drawn from a complete deck of playing cards without jokers. If the card is a club or a queen, the gambler wins double his bet. Otherwise, the dealer keeps the money. How much better are the dealer's odds in this game?

Accepted Answer

In a game where the dealer keeps a gambler's wager if he draws any card but a club or queen, his odds are 38% better than the gambler's of winning the money. There are 52 cards in a deck and 13 cards in each suit, including clubs. There are four queens in each deck, but the queen of clubs has already been counted, so only the queens of hearts, diamonds and spades count toward the gambler's list of winning cards. Therefore, the gambler needs one of 16 cards to be pulled in order to win, which make his odds of winning 31%. If his odds are 31%, the gambler's odds are the far superior 69%, which is better by 38%.

Question 17

Every five minutes a baby is born in the Nigindia. How many babies are born in a year?

Accepted Answer

There are 60 minutes in an hour, thus 60/5 minutes is 12. Multiply that by 24 hours in a day which is 288 per day. 288 x 365 days is 105,120 babies per year.
FYI, close to 4 million babies are born a day in the United States. That's one nearly every 14 seconds!
Key Takeaway: Get good working with 7, 12, 24, 30 and 60. Units of time are frequently used. Days, months, hours and minutes are frequently calculations you will have to make.

Question 18

A car that regularly sells for $29,600 has been marked down to $22,100. What is the approximate percentage of the discount?

Accepted Answer

First, determine the amount of the markdown in dollars (29,600 - 22,100 = 7,500). Then, divide the total markdown of $7,500 by the original price of $29,600, which results in 0.2533. Move the decimal point two places to the right, to determine that the markdown is a little over 25%.
Key takeaway: When working with markdowns, markups, and percentages in general, be aware of the quickest ways to calculate.
Also, remember that to change a decimal to a percent, move the decimal 2 places to the right, and add a percent sign. To change a percent to a decimal, move the decimal 2 places to the left, and remove the percent sign. If there is no decimal point, it is understood to be at the end of the number.

Question 19

What is the median for the numbers 89, 87, 94, 82, 89, 91, 95, 88, 90, 87, 92 and 93?

Accepted Answer

The median of the number set above is 89.5. The median is the number that falls in the middle of the set when the numbers are ordered by numeric value (least to greatest). The set above should be ordered as follows: 82, 87, 87, 88, 89, 89, 90, 91, 92, 93, 94, 95. When there is an even number of entrants in the set, the middle two numbers should be averaged together to find the median. This sequence has 12 numbers and the middle two are 89 and 90, which equal 89.5 when averaged.

Question 20

Kenneth receives a 6% commission on all of his real estate sales. Half of his commission goes to his broker. What is Kenneth's part of the commission on $250,000 in sales?

Accepted Answer

$7,500.
Considering what Kenneth's commission is, rather than the entire 6% commission, will make the calculation here easier. Kenneth actually gets 3% commission, after giving the broker his portion. So, our task is to multiply .03 (the decimal for 3%) x $250,000 to get $7,500.
Key Takeaway: Do all simplification that you can prior to doing the calculation. These are little things that can speed you up and give you that tiny edge.

Quiz 5: Mathematics