1

Sometimes dealing with fractions is easier than dealing with decimals, and sometimes vice versa. Here are both methods. When multiplying decimals, count how many total digits or decimal places there are to the right of the decimal point, then put the decimal point that many places to the left once you’ve done the multiplication. So (0.05)(0.5) = 0.025 (2 digits for 0.05 and 1 digit for 0.5, so 3 total digits and therefore 3 places to the left).

(5)(0.005) = 0.025

They equal the same thing, so the answer is 1. When dealing with fractions, just remember that for each decimal place, you add another zero on the bottom. So 0.05 = \(\frac{5}{100}\), because you put two zeroes on the bottom, one for each of the two digits to the right of the decimal point. Or, you could convert the decimals to fractions and then multiply:

\(\left ( \frac{5}{100} \right )\left ( \frac{5}{10} \right )=\frac{25}{1000}\)

(5)\(\left ( \frac{5}{1000} \right )=\frac{25}{1000}\)

Again, you get the same fractions on the top and bottom.

B,C, and D

It’s useful to use PITA in this problem. Note that increasing something by 10 percent is the same as multiplying by 1.10, and decreasing something by 5 percent is the same as multiplying by 0.95. Definitely use your on-screen calculator on a problem such as this.Because you are dealing with a range, it will save you time to work from the top down until you reach a correct answer, then work your way from the bottom up until you reach a correct answer.Each answer in between the smallest correct answer and the largest correct answer must also be correct.

Plug in the answers. Subtract 12 percent from each answer choice to find the one that gives you 220,000. Start with choice (C):

$193,600

$196,429

$221,200 − .12(221,200) = 194,656

$246,400 − .12(246,400) = 216,832

$250,000 − .12(250,000) = 220,000

15

We know that the difference between ⁴⁄₅ and ³⁄₄ of the container is 3 liters, so set up the following equation to solve for the volume of the container:

\(\left ( \frac{4}{5}-\frac{3}{4} \right )x=3\)

\(\left ( \frac{4}{5}-\frac{3}{4} \right )x=3\)

\(\left ( \frac{1}{20} \right )x=3\)

x = 60

Since the container is still ³⁄₄ full, it has 45 liters in it.Therefore, you’ll need to pour 15 liters back in.

Since the problem doesn’t give you prices, Plug In; since the question involves percents, use 100. If the starting price for Melange was $100, then, after the 20% decrease, the price was $80; increase that by 50%, or ¹⁄₂, and the ending price was $120. Since the ending prices for Melange and Blue were the same, the ending price for Blue was also $120. Now, work backwards. Declining by 50% is the same as being cut in half; if Blue was $120 after being cut in half, it must have been $240 before the decrease. Since that $240 was the result of a 20% increase, translate the question “240 is 120% of what” to get the equation 240 = \(\frac{120}{100}\) × x; solve for x to get the starting price for Blue, $200. Finally, now that you have both starting prices, translate the question into 20 = \(\frac{x}{100}\) × 100, and solve for x to get your answer, 200.

C and E

To solve this question, plug in the answers as the number of seniors to see if the rest of the class adds up to 150. Start with choice (C). If S = 24, then \(\frac{60}{100}\) and J = 40; since 40 = 50% of F, then F = 80; since the sum of the students is 24 + 40 + 80 = 144, which is less than 150, try larger numbers and eliminate choices (A), (B), and (C). For choice (D), if S = 25, then 25 = \(\frac{60}{100}\) J, thus there are 41.67 juniors, which is incorrect since it is impossible to have a fraction of a student. For choice (E), if S = 27, then \(\frac{60}{100}\) J, thus J = 45; since 45 = 50% of F, then F = 90; since 27 + 45 + 90 = 162, which is at least 160, choice (E) is the only correct answer.

5

First, translate the English into math.Thirty-five hundredths is 0.35 and four thousandths is 0.004. Now, do the division on your onscreen calculator; the first place after the decimal point is the tenths digit, so the answer is 5.

Sure, you could use algebra here, but most likely you won’t wind up with an answer that looks anything like choice (D). Instead, try plugging in a value for x. Make x something easy such as 1000, which makes y = 500. Plug x = 1000 into each answer choice, and it turns out that only choice (D) gives you the correct value of 500.

A,B, and E

Anything that’s a multiple of 30 will work here because of the two denominators of the two fractions in the question.Choices (C), (D), and (E) are all multiples of 30, but the remaining answers are not.You can also treat this as a PITA problem and try each answer. Take ⁵⁄₆ of each of the answer choices, then take ³⁄₅ of what’s left. Do you wind up with an integer? If so, eliminate it and keep working until you’ve tried every answer

-\(\frac{80}{81}\)

One easy way to solve this problem is to multiply both sides of the equation by 9, which gives you (w + 1) = \(\frac{9}{729}\), or (w + 1) = \(\frac{9}{729}\). Now subtract 1 from both sides to get -\(\frac{9}{729}\).