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OSAT Middle Level/Intermediate Mathematics (225) study materials
OSAT Middle Level/Intermediate Mathematics (225) study resources

OSAT Middle Level/Intermediate Mathematics (225) practice questions

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Question 1Number Sense and Operations

A student writes the prime factorization of 252 as 2^a × 3^b × 7^c. What is the value of a + b + c?

  1. 4
  2. 6
  3. 7
  4. 5 Correct answer.

Explanation

Dividing 252 repeatedly by primes gives 252 = 2 × 126 = 2 × 2 × 63 = 2^2 × 3^2 × 7. Thus a = 2, b = 2, c = 1, and the sum is 5.

Question 2Number Sense and Operations

Which number is irrational?

  1. 22/7
  2. 0.75
  3. 0.121221222... (the pattern of increasing runs of 2s continues forever) Correct answer.
  4. sqrt(49)

Explanation

A decimal that never terminates and never repeats is irrational, and the pattern of lengthening runs of 2s never becomes periodic. The other three values are a rational fraction, a terminating decimal, and the integer 7.

Question 3Number Sense and Operations

Simplify (3 × 10^5)(4 × 10^-8) and express the result in scientific notation.

  1. 12 × 10^-3
  2. 1.2 × 10^-2 Correct answer.
  3. 1.2 × 10^-3
  4. 1.2 × 10^-4

Explanation

Multiply the coefficients: 3 × 4 = 12, and add exponents: 10^5 × 10^-8 = 10^-3. Then 12 × 10^-3 = 1.2 × 10^-2 in proper scientific notation.

Question 4Number Sense and Operations

A jacket priced at $80 is discounted 25%, and then an additional 10% is taken off the reduced price. What is the final price?

  1. $60.00
  2. $54.00 Correct answer.
  3. $52.00
  4. $56.00

Explanation

After the 25% discount the price is 0.75 × 80 = $60. The second discount applies to $60, giving 0.90 × 60 = $54. Successive percent discounts multiply rather than add.

Question 5Number Sense and Operations

What is the greatest common factor of 84 and 126?

  1. 6
  2. 14
  3. 42 Correct answer.
  4. 21

Explanation

Factor both numbers: 84 = 2^2 × 3 × 7 and 126 = 2 × 3^2 × 7. The GCF takes the lowest power of each shared prime, giving 2 × 3 × 7 = 42.

Question 6Number Sense and Operations

Evaluate 5 + 2 × (3^2 - 4) ÷ 2 using the standard order of operations.

  1. 10 Correct answer.
  2. 7
  3. 13
  4. 21

Explanation

First evaluate the exponent and parentheses: 3^2 - 4 = 5. Then multiply and divide left to right: 2 × 5 ÷ 2 = 5. Finally 5 + 5 = 10.

Question 7Number Sense and Operations

Which expression is equivalent to 3/4 ÷ 2/5?

  1. 6/20
  2. 8/15
  3. 15/8 Correct answer.
  4. 3/10

Explanation

Dividing by a fraction means multiplying by its reciprocal: 3/4 × 5/2 = 15/8. The result is greater than 1 because 3/4 exceeds 2/5.

Question 8Number Sense and Operations

The number 0.000406 written in scientific notation is:

  1. 4.06 × 10^-4 Correct answer.
  2. 40.6 × 10^-5
  3. 4.06 × 10^4
  4. 4.06 × 10^-5

Explanation

Move the decimal point four places right to get 4.06, so the exponent is -4. Checking: 4.06 × 10^-4 = 0.000406.

Question 9Number Sense and Operations

A recipe uses 3 cups of flour for every 5 cups of milk. If a cook uses 18 cups of flour, how many cups of milk are needed to keep the ratio?

  1. 30 Correct answer.
  2. 10.8
  3. 27
  4. 24

Explanation

Set up the proportion 3/5 = 18/x. Cross-multiplying gives 3x = 90, so x = 30. The flour amount was multiplied by 6, so the milk amount must also be multiplied by 6.

Question 10Number Sense and Operations

Which statement about the set of integers is true?

  1. It contains no negative numbers.
  2. It is closed under taking square roots.
  3. It is closed under division.
  4. It is closed under subtraction. Correct answer.

Explanation

The difference of any two integers is always an integer, so the set is closed under subtraction. Division, square roots, and negatives all produce results outside the integers in some cases.

Question 11Number Sense and Operations

A store marks up a wholesale item by 40% and later offers a 20% discount on the marked-up price. Compared with the wholesale price, the final price is:

  1. equal
  2. 20% higher
  3. 8% higher
  4. 12% higher Correct answer.

Explanation

Multiply the factors: 1.40 × 0.80 = 1.12, so the final price is 112% of the wholesale price, a 12% increase. Percent changes compound multiplicatively, not additively.

Question 12Number Sense and Operations

What is the least common multiple of 18 and 24?

  1. 432
  2. 144
  3. 6
  4. 72 Correct answer.

Explanation

Write 18 = 2 × 3^2 and 24 = 2^3 × 3. The LCM takes the highest power of each prime: 2^3 × 3^2 = 72.

Question 13Number Sense and Operations

Simplify the expression (2^3 × 2^-5) ÷ 2^-4.

  1. 1/16
  2. 4 Correct answer.
  3. 1/4
  4. 16

Explanation

Combine the numerator: 2^3 × 2^-5 = 2^-2. Then divide by subtracting the exponent: 2^-2 ÷ 2^-4 = 2^(-2 + 4) = 2^2 = 4.

Question 14Number Sense and Operations

A number is multiplied by 0.6 and the result is 27. What is the original number?

  1. 45 Correct answer.
  2. 162
  3. 16.2
  4. 4.5

Explanation

Solve 0.6x = 27 by dividing both sides by 0.6, giving x = 45. Checking: 0.6 × 45 = 27.

Question 15Number Sense and Operations

Which list orders the numbers from least to greatest? 7/8, 0.85, 5/6, 0.9

  1. 5/6, 0.85, 7/8, 0.9 Correct answer.
  2. 0.85, 5/6, 0.9, 7/8
  3. 5/6, 7/8, 0.85, 0.9
  4. 0.85, 5/6, 7/8, 0.9

Explanation

Convert to decimals: 5/6 ≈ 0.833, 0.85 = 0.85, 7/8 = 0.875, and 0.9 = 0.9. Ordering these gives 5/6, 0.85, 7/8, 0.9.

Question 16Number Sense and Operations

Which value is equal to 0.454545... (the block 45 repeats forever)?

  1. 4/9
  2. 5/11 Correct answer.
  3. 45/100
  4. 9/20

Explanation

Let x = 0.454545... Then 100x = 45.4545..., and subtracting gives 99x = 45, so x = 45/99 = 5/11.

Question 17Number Sense and Operations

A student claims that the sum of any two irrational numbers is irrational. Which pair of numbers shows the claim is false?

  1. sqrt(2) and -sqrt(2) Correct answer.
  2. sqrt(5) and 1
  3. sqrt(2) and sqrt(3)
  4. pi and pi

Explanation

The sum sqrt(2) + (-sqrt(2)) equals 0, which is rational, so the claim fails. The other pairs have irrational sums, which does not disprove the claim but also does not test it.

Question 18Number Sense and Operations

Convert the repeating decimal 2.363636... to a fraction in simplest form.

  1. 236/99
  2. 26/11 Correct answer.
  3. 13/5
  4. 118/50

Explanation

Let x = 2.363636... Then 100x = 236.3636..., and subtracting gives 99x = 234, so x = 234/99 = 26/11.

Question 19Number Sense and Operations

A quantity increases by 20% each year for two years, then decreases by 20% in the third year. Compared with the starting value, the final value is:

  1. 4% lower
  2. 20% higher
  3. equal
  4. 15.2% higher Correct answer.

Explanation

Apply the factors: 1.2 × 1.2 × 0.8 = 1.152, so the final value is 115.2% of the start, a 15.2% increase. Percent changes compound, so a later decrease does not undo the earlier gains.

Question 20Number Sense and Operations

If n is a positive integer and 2^n divides 12! but 2^(n+1) does not, what is n?

  1. 10 Correct answer.
  2. 8
  3. 6
  4. 12

Explanation

Count factors of 2 in 12!: there are floor(12/2) = 6, floor(12/4) = 3, and floor(12/8) = 1, totaling 10. So the highest power of 2 dividing 12! is 2^10.

Question 21Number Sense and Operations

Which expression has the greatest value?

  1. (2/3)^0
  2. (2/3)^-2 Correct answer.
  3. (2/3)^2
  4. (2/3)^-1

Explanation

A negative exponent takes the reciprocal: (2/3)^-2 = (3/2)^2 = 9/4 = 2.25. The others are 4/9, 1, and 3/2, all smaller than 2.25.

Question 22Number Sense and Operations

A number has exactly three positive divisors. Which number satisfies this condition?

  1. 49 Correct answer.
  2. 15
  3. 12
  4. 8

Explanation

A number has exactly three divisors only when it is the square of a prime. Since 49 = 7^2, its divisors are 1, 7, and 49.

Question 23Number Sense and Operations

A car travels 150 miles on 6 gallons of fuel. At this rate, how many gallons are needed for 275 miles?

  1. 11 Correct answer.
  2. 9.5
  3. 12.5
  4. 10

Explanation

The rate is 150 ÷ 6 = 25 miles per gallon. Then 275 ÷ 25 = 11 gallons. Alternatively, solve the proportion 150/6 = 275/x.

Question 24Number Sense and Operations

What is the value of the digit 7 in the number 3,472.58?

  1. 7 tenths
  2. 7 thousandths
  3. 7 hundreds
  4. 7 tens Correct answer.

Explanation

In 3,472.58 the digits to the left of the decimal are 4 in the hundreds place, 7 in the tens place, and 2 in the ones place. So the 7 represents 7 tens, or 70.

Question 25Number Sense and Operations

Which expression is equivalent to 0.00032 ÷ 0.008?

  1. 0.4
  2. 4
  3. 0.04 Correct answer.
  4. 0.004

Explanation

Rewrite both in scientific notation: 3.2 × 10^-4 divided by 8 × 10^-3 equals 0.4 × 10^-1 = 0.04. Equivalently, 32 ÷ 800 = 0.04.

Question 26Number Sense and Operations

A student evaluates 48 ÷ 6 × 2 by first multiplying 6 × 2 and then dividing 48 by 12, obtaining 4. What is the correct value of the expression, and why?

  1. 16, because the multiplication sign binds more tightly than the division sign
  2. 4, because multiplication is performed before division
  3. 16, because multiplication and division are performed left to right Correct answer.
  4. 96, because division is performed before multiplication

Explanation

In the standard order of operations, multiplication and division have equal priority and are evaluated from left to right. Therefore 48 ÷ 6 is computed first to get 8, and then 8 × 2 = 16. Grouping the 6 × 2 first changes the meaning of the expression.

Question 27Number Sense and Operations

A number is multiplied by 0.25 and the result is 9. What is the original number?

  1. 13
  2. 4
  3. 36 Correct answer.
  4. 2.25

Explanation

Multiplying by 0.25 is the same as dividing by 4, so the original number is 9 × 4 = 36. Equivalently, solve 0.25x = 9 by dividing both sides by 0.25.

Question 28Number Sense and Operations

Which statement about the set of rational numbers is true?

  1. The product of a rational number and an irrational number is always irrational.
  2. Every rational number can be written as a terminating decimal.
  3. Every integer is a rational number, but not every rational number is an integer. Correct answer.
  4. The sum of two rational numbers is always rational.

Explanation

Every integer n can be written as n/1, so integers are rational, but numbers such as 1/2 are rational and not integers. The other statements fail: 1/3 has a repeating decimal, and 0 × √2 = 0 is rational.

Question 29Number Sense and Operations

A shirt originally priced at $40 is discounted by 15%. What is the sale price?

  1. $25
  2. $6
  3. $46
  4. $34 Correct answer.

Explanation

A 15% discount reduces the price to 85% of the original, so the sale price is 0.85 × $40 = $34. Equivalently, the discount is 0.15 × $40 = $6, and $40 − $6 = $34.

Question 30Number Sense and Operations

Which number is equivalent to 3/8?

  1. 2.666...
  2. 0.83
  3. 0.375 Correct answer.
  4. 0.38

Explanation

Dividing 3 by 8 gives 0.375 exactly. The other choices either misread the fraction as a decimal or invert the numerator and denominator.

Question 31Number Sense and Operations

Which expression is equivalent to 5^3 × 5^4?

  1. 25^7
  2. 5^12
  3. 25^12
  4. 5^7 Correct answer.

Explanation

When multiplying powers with the same base, add the exponents: 5^3 × 5^4 = 5^(3+4) = 5^7. The base stays 5, and the exponents are added, not multiplied.

Question 32Number Sense and Operations

Which list orders the numbers 0.6, 2/3, 0.65, and 5/8 from least to greatest?

  1. 2/3, 0.65, 5/8, 0.6
  2. 0.6, 5/8, 0.65, 2/3 Correct answer.
  3. 0.6, 0.65, 5/8, 2/3
  4. 5/8, 0.6, 2/3, 0.65

Explanation

Writing all numbers as decimals gives 0.6, 0.625, 0.65, and 0.666..., so the order is 0.6 < 5/8 < 0.65 < 2/3. Comparing common denominators or decimal equivalents confirms this order.

Question 33Number Sense and Operations

A recipe calls for 2 cups of sugar for every 3 cups of flour. If a baker uses 15 cups of flour, how many cups of sugar are needed?

  1. 10 Correct answer.
  2. 12
  3. 7.5
  4. 22.5

Explanation

The ratio of sugar to flour is 2:3, so set up 2/3 = x/15. Cross-multiplying gives 3x = 30, so x = 10 cups of sugar. The flour amount is five times 3 cups, so the sugar is five times 2 cups.

Question 34Number Sense and Operations

What is the value of the digit 4 in the number 27.348?

  1. 4 tenths
  2. 4 tens
  3. 4 thousandths
  4. 4 hundredths Correct answer.

Explanation

In 27.348, the digits after the decimal point are 3 tenths, 4 hundredths, and 8 thousandths. Therefore the 4 is in the hundredths place and represents 4/100.

Question 35Number Sense and Operations

A number is multiplied by 1.25 and the result is 45. What is the original number?

  1. 43.75
  2. 33.75
  3. 56.25
  4. 36 Correct answer.

Explanation

Solve 1.25x = 45 by dividing both sides by 1.25: x = 45 ÷ 1.25 = 36. Multiplying 36 by 1.25 gives 45, confirming the result.

Question 36Number Sense and Operations

Which expression is equivalent to 0.00045 ÷ 0.0009?

  1. 5
  2. 0.05
  3. 0.5 Correct answer.
  4. 50

Explanation

Multiply numerator and denominator by 100,000 to clear the decimals: 45 ÷ 90 = 0.5. Alternatively, 0.00045 is half of 0.0009, so the quotient is 0.5.

Question 37Number Sense and Operations

Which of these numbers is irrational?

  1. √16
  2. 22/7
  3. 0.121212...
  4. √7 Correct answer.

Explanation

An irrational number cannot be written as a ratio of integers. Since 7 is not a perfect square, √7 has a nonterminating, nonrepeating decimal expansion and is irrational. The other choices simplify to rational numbers.

Question 38Number Sense and Operations

A jacket is marked up 25% from its wholesale price and later sold at a 20% discount on the marked-up price. Compared with the wholesale price, the final selling price is

  1. 5% lower
  2. 5% higher
  3. 25% higher
  4. the same Correct answer.

Explanation

After a 25% markup, the price is 1.25 times the wholesale price. Applying a 20% discount multiplies that by 0.80, giving 1.25 × 0.80 = 1.00 times the wholesale price, so the final price equals the wholesale price.

Question 39Number Sense and Operations

A quantity decreases by 10% each year for two years. Compared with the starting amount, the total decrease is

  1. 20%
  2. 19% Correct answer.
  3. 18%
  4. 10%

Explanation

After two years the amount is 0.90 × 0.90 = 0.81 of the original, so the decrease is 1 − 0.81 = 0.19, or 19%. Successive percent changes multiply rather than add.

Question 40Number Sense and Operations

A car travels 180 miles on 6 gallons of fuel. At this rate, how many gallons are needed for 270 miles?

  1. 12
  2. 9 Correct answer.
  3. 7.5
  4. 10

Explanation

The car travels 180 ÷ 6 = 30 miles per gallon. To travel 270 miles requires 270 ÷ 30 = 9 gallons. Equivalently, 270 is 1.5 times 180, so the fuel needed is 1.5 × 6 = 9 gallons.

Question 41Number Sense and Operations

What is the least common multiple of 12 and 18?

  1. 72
  2. 216
  3. 36 Correct answer.
  4. 6

Explanation

The prime factorizations are 12 = 2² × 3 and 18 = 2 × 3². Taking the highest power of each prime gives 2² × 3² = 36, the least common multiple.

Question 42Number Sense and Operations

A student simplifies (3^4 × 3^−2) ÷ 3^−3. What is the simplified value?

  1. 3^9
  2. 3^−1
  3. 3^1
  4. 3^5 Correct answer.

Explanation

In the numerator, 3^4 × 3^−2 = 3^(4−2) = 3^2. Dividing by 3^−3 subtracts the exponent: 3^2 ÷ 3^−3 = 3^(2−(−3)) = 3^5.

Question 43Number Sense and Operations

Which value is equal to 0.272727... (the block 27 repeats forever)?

  1. 27/99
  2. 2/7
  3. 3/11 Correct answer.
  4. 27/100

Explanation

Let x = 0.272727... Then 100x = 27.272727..., and subtracting gives 99x = 27, so x = 27/99 = 3/11. The repeating block has two digits, so the denominator is 99 before simplifying.

Question 44Number Sense and Operations

A number is increased by 50% and then decreased by 50%. Compared with the original number, the final result is

  1. 50% less
  2. 25% more
  3. 25% less Correct answer.
  4. unchanged

Explanation

Increasing by 50% multiplies by 1.5, and decreasing by 50% multiplies by 0.5. The combined factor is 1.5 × 0.5 = 0.75, so the final result is 75% of the original, which is 25% less.

Question 45Number Sense and Operations

If n is a positive integer and 3^n divides 15! but 3^(n+1) does not, what is n?

  1. 4
  2. 6 Correct answer.
  3. 7
  4. 5

Explanation

Count the factors of 3 in 15!: multiples of 3 contribute 5 (3, 6, 9, 12, 15), and multiples of 9 contribute one extra each (9), for a total of 6. Thus the highest power of 3 dividing 15! is 3^6, so n = 6.

Question 46Number Sense and Operations

A student claims that the product of a nonzero rational number and an irrational number is always irrational. Which pair of numbers shows that the claim is false?

  1. 3 and √3
  2. 0 and √2 Correct answer.
  3. 2 and √2
  4. 1/2 and √2

Explanation

The product 0 × √2 equals 0, which is rational. The claim excludes zero by specifying a nonzero rational number, but 0 is rational, so this pair demonstrates that a rational times an irrational can be rational.

Question 47Number Sense and Operations

The number 0.000305 written in scientific notation is

  1. 3.05 × 10^−4 Correct answer.
  2. 30.5 × 10^−5
  3. 3.05 × 10^4
  4. 3.05 × 10^−3

Explanation

Move the decimal point four places to the right to obtain 3.05, so the exponent is −4. Thus 0.000305 = 3.05 × 10^−4.

Question 48Number Sense and Operations

A store applies a 6% sales tax to a purchase and then applies a 10% loyalty discount to the taxed total. Which single multiplier applied to the original price gives the same final cost?

  1. 0.954 Correct answer.
  2. 0.900
  3. 0.960
  4. 1.166

Explanation

The tax multiplies the price by 1.06, and the discount then multiplies that result by 0.90. Composing the two operations gives 1.06 × 0.90 = 0.954, so a single multiplier of 0.954 reproduces the final cost.

Question 49Number Sense and Operations

A student writes the repeating decimal 0.4̄5̄ (the two-digit block 45 repeats forever) as a fraction in lowest terms. Which fraction is correct?

  1. 45/100
  2. 15/33
  3. 5/11 Correct answer.
  4. 9/20

Explanation

Let x = 0.454545..., then 100x = 45.454545..., and subtracting gives 99x = 45, so x = 45/99 = 5/11. Dividing numerator and denominator by 9 reduces the fraction to lowest terms.

Question 50Number Sense and Operations

For a positive integer n, let v(n) denote the exponent of 2 in the prime factorization of n. What is v(20!) − v(10!)?

  1. 8
  2. 12
  3. 10 Correct answer.
  4. 18

Explanation

By Legendre's formula, v(20!) = 10 + 5 + 2 + 1 = 18 and v(10!) = 5 + 2 + 1 = 8. The difference is 18 − 8 = 10, which also matches the count of factors of 2 contributed by the integers 11 through 20.