Explanation:
Strategy: work with the answers. Go through the answers, substituting each
for x and using it in the problem until it works. For example, to check x = 4
(answer B), subtract 4 from 36 to get 32, and divide by 4, resulting in 8, so answer
A is incorrect. You will find that only answer D gives you the final result of 2.
Math Teacher Solution: Convert the words into an equation for x:
36 − x
x = 2.
Multiplying both sides by x gives 36 − x = 2x so that x = 12.
Explanation:
Correct answer 9
Explanation:
One way to do this is to just start listing numbers di-visible by 7, say, and checking to see which of those are
divisible by both 3 and 5. Some time later, you’ll realizethat no positive integer (that means greater than zero)
which is less than 100 can be divided by all three given numbers.
Faster is to realize that the three numbers given are all prime, so that the smallest number divisible by all three
(the least common multiple) is 3 × 5 × 7 = 120
Explanation:
Since 0.6666 = 2/3, and any fraction with integers on the top and bottom is a rational number, answer B is correct.
Why are the other answers incorrect? Thenumber 2/3 can be correctly gridded as “.666” or “.667” or “2/3” but not
as “0.66”. Also, 67/100 = 0.67 is not the same as 0.6666. Finally, 0.6666 isn’t a little devil every night of the week,
so we can’t be sure about last night.
Explanation:
Since the greatest negative integer is −1, the answer is 2(−1) + 8 = 6.
Explanation:
Strategy: plug in real numbers. Try easy prime numbers for m and n and plug them in, checking to see which of the
answers is prime. For example, let m = 3 and n = 5. For this choice, none of the answers is prime! When you are plugging
in numbers, you need to be able to eliminate all answers except for one. If you can only eliminate a few answers, or none
at all, try different numbers to plug in.However, if you do eliminate some answers, you don’t need to check those particular
answers again. For this question, if you try m = 3 and n = 7 next, you will find that only answer B is prime, so it must be correct.
Math Teacher Solution: If m and n are prime numbers, then answer B is a composite number and can’t be prime. Answers A and
C are even: since m and n are odd numbers, m + n is even, and m + n + 2 is even as well. Also, mn is odd, so that mn + 1 (answer D)
is even. By process of elimination, answer E is correct.
Explanation:
Strategy: work with the answers. Go through the answers until you find one that works.
For example, if Aubrey bought 4 cabbages, then she paid $48, leaving $104 − $48 = $56 for Jack,
who would have bought 7 tomatoes since $56/$8 = 7. But, 4 + 7 = 11, not 10, so answer D is incorrect.
Continuing in this way, you will find that only answer B works.
Math Teacher Solution: Let t be the number of tomatoes and B be the number of cabbages. Set up two
equations: 8t+12c = 104 and t+c = 10. Solve for t and c by solving one equation for one variable and substituting
into the other equation. Here,
t = 10−c so that 8(10−c)+12c = 104.
Simplifying, 80−8c+12c = 80+4c = 104
so that B = 6.
Explanation:
Remember that “between” means that you should not include the end points. So,
the even integers between −10 and 10 are −8, −6, −4, −2, 0, 2, 4, 6, and 8.
Explanation:
Strategy: plug in real numbers. Suppose 5 sandwiches were ordered (i.e., plug in 5 for c. These sandwiches would cost
$8 each, or $40 in all. With the fee, the total cost would be $50 + $40 = $90. Go through the answers, plugging in 5 for c
until you get $90. This occurs only for answer B.
Math Teacher Solution: If $8 is the cost per sandwich (excluding the fee), then 8c is the cost in dollars for c sandwiches.
Since the fee is a one-time charge, the total cost in dollars is just 50 + 8c.
Explanation:
Strategy: plug in real numbers. Try using a real number in place of y. As always, use a number that makes life easy for yourself.
In this case, since 30 erasers costs y dollars, we will try setting y equal to 30 dollars. In this way, each eraser costs 1 dollar, and
70 erasers will cost 70 dollars. Go through the answers putting in 30 for y until you get an answer of 70. This occurs only for
answer E.
Math Teacher Solution: Let x be the total cost of 70 erasers. Then we set up a proportion:
30 = 70
y x
and solve for x by cross-multiplying. This gives: 30x = 70y so that x = 7y/3.
Explanation:
Strategy: work with the answers. Use the answers to try choices for the number of green marshmallows. If there are
9 green marshmallows (answer C), then there are 6 + 9 = 15 total marshmallows, so the probability of choosing a
purple one is 6/15 = 2/5 > 1/3, so answer C is incorrect. We need more green marshmallows to get a lower probability
for choosing a purple marshmallow, so we try an answer to the right. If there are 12 green marshmallows (answer D),
then there are 6 + 12 = 18 total marshmallows, so the probability of choosing a purple one is 6/18 = 1/3. Answer D is correct.
Math Teacher Solution: Let x be the number of green marshmallows. Then, 6 + x is the total number of marshmallows. The
probability of choosing a purple marshmallow is the number of purple marshmallows (6) divided by the total number
of marshmallows (6 + x). So, we need:
6/6+x = 1/3
Cross-multiplying, 18 = 6 + x so that x = 12.
Explanation:
Correct answer 24
Explanation:
The smallest prime number is 2, and the largest prime number less than 10 is 7, so
the answer is 2 + 7 = 9.
Explanation:
Plug in a prime number for p, and see which of the answers is also prime. (Hint: memorize the first few primes.
You will probably want to try more than one prime number for this question.) Since there is only one
correct answer, four of the answers given can never be prime numbers when p is prime. You’ll see that answer
A is the only one that can result in more prime numbers. How to do this problem without plugging in numbers?
Notice that answer E isn’t even an integer (unless p = 2, but then p/2 = 1 isn’t prime). Answers B and C are
always even and greater than 2, so these answers can never be prime. Answer d can always be divided by p,
so it is never prime. The only answer remaining is A, so it must be correct.
Explanation:
The smallest prime number is 2, and the largest negative even integer is −2, so the
answer is 0.
Explanation:
Strategy: plug in real numbers. Pick a number to plug in for m which, when divided by 7, leaves a remainder of 4.
A good choice might be 11, but you could have also chosen 18, 25, etc. Now use the number that you chose: add 26
to it, and see what the remainder is when you divide the new number by 7. When 11+26 = 37 is divided by 7, the remainder
is 2, so choice B is correct.
Math Teacher Solution: Since the remainder is 4 when m is divided by 7, we can write m = 7i + 4 for some integer i.
For example, if i = 1, then m = 11; if
i = 2, then m = 18, and so forth. Then, m + 26 = 7i + 30 = 7i + 7 · 4 + 2 so that
m + 26 = 7(i + 4) + 2. So, the remainder is 2.
Explanation:
Correct answer: 2
Explanation:
Correct answer 6
Explanation:
15% of 180 = 15/100 × 180 = 27
Let 𝑥 be the number then, 𝑥 = 27 + 12 = 39
Explanation:
The area of the square is 64 inches. Therefore, the side of the square is the square root of the area.
(\sqrt{64} = 8\) inches
Four times the size of the square is the perimeter:
4 × 8 = 32 inches
Explanation:
First, find the number.
Let 𝑥 be the number. Write the equation and solve for 𝑥.
150% of a number is 75, then:
1.5 × 𝑥 = 75 ⇒ 𝑥 = 75 ÷ 1.5 = 50
90 of 50 is:
0.9 × 50 = 45
Explanation:
Find the difference of each pair of numbers:
2, 3, 5, 8, 12, 17, 23, _, 38
The difference of 2 and 3 is 1, 3 and 5 is 2, 5 and 8 is 3, 8 and 12 is 4, 12 and 17 is 5, 17
and 23 is 6, 23 and next number should be 7.
The number is 23+7=30
Explanation:
Let 𝑥 be the number of shoes the team can purchase. Therefore, the team can purchase 120𝑥.
The team had $20,000 and spent $14000. Now the team can spend on new shoes $6000 at most.
Now, write the inequality:
120𝑥+14,000≤20,000
Explanation:
Prime factorizing of 70 = 2×5×7
Prime factorizing of 66 = 2×3×11
GCF = 2
7- C
12 = 0.5
79 = 0.77
65% = 0.65
Explanation:
Number of books in 30% of red box =30100×30=9→30−9=21
