Annotated Answers to Math Competency -B
by Bob Stepno
(former English major)

1. Finding five percent less: five percent  means .05 times the old total.  Less  means subtraction. Take the old total, multiply it by .05, then subtract the result from the old total. 8,300 times .05 is 415 and 8,300 - 415 is 7,885.

2. Pretty easy, but confusing because the amount of the decrease and the pay rate after the decrease are both $4.02. It's obvious they cut his pay in half, and half is 50 percent. To do it the long way, divide the absolute value of the decrease, 4.02, by the base, 8.04. Same answer.

3. Maybe Tom Sawyer wouldn’t do it this way, but this question specifies that you’re only painting one side of the fence. Square footage is length times width; 2,100 square feet in this case. You know one gallon will cover 400, so divide 2,100 by 400 to find out that you need 5.25 gallons.

4. If he chose the third through ninth, he essentially removed the first, second, tenth, eleventh and twelfth. "Remove" and "subtract" are the same thing. So subtract those five students from the dozen total, and you have seven. Or just hold up your thumb and say "three," your first finger and say "four" and keep going until you hit nine. Then count the fingers.

5. If your two-year-old was born in December 1994 and just had a second birthday in 1996, you would subtract 1994 from 1996 to get her age. The same is true for old Bruce. Subtract 1949 from 2016, and don’t forget to carry the one. The answer is 67.

6. Per cent is per hundred, so the relationship of one to 12 is like the proportion of something out of one hundred. Divide 100 by 12 to get the answer. If you need to see it in algebra, it looks like this:

   1           X
  —   =    —
  12       100

 Multiply both sides by 100 to get rid of the fractions. 1/12 times 100 is 100/12, so the answer is 8 and a fraction.

7. This time you know how many square feet you need to pay for, and you know how much one square foot costs, so you can just multiply the cost of one by the total:  2500 x 65 = $162,500

8. After you eat the quarter pizza, three quarters are left. Imagine the whole pizza as 16 slices, of which you get four. The other 12 will be divided by four roommates, or three slices each. Those three slices would be 3/16 of the whole pie. If you assume the pizza has a different number of pieces, you’ll get the same answer after a little more arithmetic. (With a 12-slice pizza your roommates get 9 slices to split four ways, which is 9/4 or 2.25 slices each, or 2.25/12 of the whole pie. Multiplying by four to get rid of the decimal gives us 9/48, which reduces to 3/16 again.)

9. Calling it a Gwamix ruler means that it shows 12 Gwamixes. That is, each big mark with a number indicates a whole Gwamix. Count the spaces defined by the big and little marks: there are five spaces in each Gwamix, so each mark indicates 1/5 of a Gwamix.

10. For simplicity, think of this as 10 work days. At 50 percent probability, you'd be late on five of them. Of those five late days, the probability of the boss noticing is 50 percent again, or 2.5 days on which you might be late and noticed. Since 2.5 is one quarter of the ten days, it's easy to recognize as 25 percent of 10, just like a quarter is 25 percent of a dollar.

11. If 37 out of 217 failed, subtraction tells you that 180 passed. The base is all students taking the test or 217, and the base goes at the bottom of the fraction; that is, 180 ÷ 217, which equals .829 or 83 percent.

12. If you drop your metric phobia, this one may sound familiar. You know the value of the old measure for one thing and you know how many of the new measure you have.  At 0.454 kilograms to a pound, a pound is a little less than half a kilogram. So the number for weight in pounds will be a little more than double the number for weight in kilograms. In algebra, the problem is 0.454x=43, and you divide both sides by 0.454 to get the answer x= 94.7.

13. The two margins total one inch, leaving 7 ½ inches of the page. The three pictures need two spaces between them or a total  of ½ inch, which leaves 7 inches of the page to work with. To fit the three pictures, divide 7 by 3. The answer is 2.33.

14. Similar to the mural problem, but with no outer margin. Six pens means five one-foot spaces between them. So subtract five from 78, then divide by 6. (73/6=12 1/6) Since feet and inches aren’t decimals, leave this as a fraction:   1/6 of 12 inches is 2 inches.

15. Remember the word  ratio  from school? Seven is to twelve as X is to 18. Put it in fractions:

   7         X
  —   =  —
  12       18

Then do the cross-multiplication trick you learned in high school. If the fractions are equal, so are the multiples of the diagonals. Multiple seven by 18 and divide by 12 or X =  (7 * 18)/12. The answer is 10.5 inches.

16. First find the increase in dollars: 7,300 - 5,500 = 1,800. The old computer budget is the base. Remember that the base always goes at the bottom of the fraction: 1,800 ÷ 5,500 = .327 or 32.7 percent.

17. Estimating the distance between the Old Well and Carroll Hall doesn’t require a tape measure, as long as you can visualize a picture of campus. The solution assumes that you remember that a mile is 5,000-and-something feet and that you can visualize 100 yards (300 feet) from your past experience with football or other sports. The two landmarks are more than one football field apart, but they certainly don’t approach a mile.  The correct answer is 500 feet.

18. The sun rises in the East, so it gets to Washington first and marches toward California, leaving D.C. in the dark while it is still light ("earlier") in California. So when it is midnight in Los Angeles it is an hour "later" in the Rockies, another hour later in the Midwest, and another hour later in Washington (3 a.m.).

19. Hard times in Goochville if you tried to solve this with arithmetic... "Median" is the point above and below which half of the cases fall -- after you put them in order of size. Two cases are above $12,500; two are below. Easy. If you added them up and divided by five, you found the "mean" and marked "none of the above."

20. On the Dakota table, notice which directions add up to 100 percent: the columns. So the percentages are OF the columns. The column headings are "North" and "South," so the only true statements about percentages will be in the form "something percent of North Dakotans" or "something percent of South Dakotans."

21. Assuming that you don't keep demographic statistics in your head, this is about making a ballpark guess and eliminating the incorrect answers. Start with what you know. You probably know a few cities of over 100,000 in your state, or any state you can think of --  but you probably don't know 20 of them. Even 20 would be 2 million people per state, or 100 million for the 50 states. That doesn't even count the really big and really small communities. So the 24 million total is way too low. And there are a lot of smaller communities in most states, and a few much larger ones in some states. An old TV series helps me, since it started with an announcer saying there were "8 million stories in the naked city," give or take a million or two. Not every state has a city that big, but let's pretend they all did have at least one city. Even that exaggeration would give the country only 500 million people -- a half billion. The population would have to be almost five  times that size to be 2.4 billion, so the over-a-billion answers must be wrong, and 240 million must be the closest to reality.

22. There’s that magic word  per  again, which means you have to divide something. Divide the total valuation by the $100 to find out how many taxable units there were:  1,275. Then divide the tax total by those units of value. That is, $1,466.25/1,275. Be careful of the decimal point; just looking at the problem you should see that the answer is between $1 and $2.

23.  Just remember Stanley Kubrick's movie 2001. Or think of the first century as the years 1 through 100 (there was no year zero), the second as 101 through 200... the nineteenth as 1801 through 1900, and so on.

24. It helps to remember that  percent  means  per hundred. So five percent is .05. If the total has grown by five percent since 1982, today’s total is 105 percent of (1.05 times) the 1982 total. In algebra, you’d say 1.05X = 693. In English, you’d ask, “What number when multiplied by 1.05 equals 693? To find out, just divide 693 by 1.05 and get 660.

25. What does "two times more" mean? The expression is ambiguous at best, and you shouldn’t use it. One hundred percent is one "time." So two times more would be a 200 percent increase or three times as many. Six is a 100 percent increase over  three and therefore two times as many, not two times “more.”

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