# ISEE Math Practice Test

#### The result of subtracting a number x from 36 and dividing the difference by x is 2. What does x represent?

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.

#### Angela has a 6-mile-per-hour running speed. How many miles can she run in 90 minutes if she keeps the same pace?

Explanation:

Correct answer 9

#### How many positive integers under 100 are divisible by three, five, and seven?

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

#### Which of the following statements regarding 0.6666 is correct?

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.

#### What number is eight times the largest negative integer?

Explanation:

Since the greatest negative integer is −1, the answer is 2(−1) + 8 = 6.

#### What is the product of the four largest negative integers and the smallest positive integer?

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.

#### Jack purchased purple tomatoes for $8 apiece from a premium vegetable stall, and Carol purchased organic huge cabbages for $12 each. They spent a total of $104 on ten vegetables. Carol bought how many cabbages?

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.

#### Between the numbers -10 and 10, how many even integers are there?

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.

#### For a one-time price of $50 + $8 per sandwich, a caterer will prepare cabbage sandwiches for special events. In dollars, how much does an order of c cabbage sandwiches cost?

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.

#### A total of x dollars will be spent on 30 identical erasers. What is the total cost of 70 erasers in terms of x at this rate?

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.

#### Six purple marshmallows are included in a carton of only purple and green marshmallows. How many green marshmallows are in the box if the likelihood of selecting a purple marshmallow from the box is 1/3?

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.

#### When a number is divided by four. The result is divided by three, resulting in a final score of two. What was the original number?

Explanation:

Correct answer 24

#### What is the sum of the smallest and greatest primes that is less than ten?

Explanation:

The smallest prime number is 2, and the largest prime number less than 10 is 7, so

the answer is 2 + 7 = 9.

#### Which of the following might be a prime number if p is a prime number?

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.

#### What is the sum of the lowest prime and the largest negative even integer?

Explanation:

The smallest prime number is 2, and the largest negative even integer is −2, so the

answer is 0.

#### The remainder of dividing the positive integer m by 7 is 4. What is the remainder after dividing m + 26 by 7?

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.

#### In one hour, a farmer can collect 12 cabbages. How many hours would it take two farmers to gather 48 cabbages if they worked at the same rate?

Explanation:

Correct answer: 2

#### A specific rectangle has a surface area of 72. What is the width of the rectangle if the length of the rectangle is twice the width?

Explanation:

Correct answer 6

#### In the following equation, what is the value of x?

10+4(𝑥+5−5𝑥) = 30

Explanation:

15% of 180 = 15/100 × 180 = 27

Let 𝑥 be the number then, 𝑥 = 27 + 12 = 39

#### What is the perimeter of a square with a 64-square-inch surface area?

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

#### What is the 90 percent of a number if 150 of a number equals 75?

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

#### In the given numbers, what is the missing term?

\( 2, 3, 5, 8, 12, 17, 23, _, 38\)

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

#### A football squad had a budget of $20,000 for equipment. A new ball cost the squad $14,000. Each pair of new athletic sneakers is $120. Which of the following inequities best describes the team's ability to buy new shoes?

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

#### What is the greatest common factor between the numbers 66 and 70?

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

#### The size of the red box is 30% larger than the size of the blue box. How many books are there in the blue box if there are 30 in the red box?

Explanation:

Number of books in 30% of red box =30100×30=9→30−9=21