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WEST-E Middle Level Mathematics (NES 203) study materials
WEST-E Middle Level Mathematics (NES 203) study resources

WEST-E Middle Level Mathematics (NES 203) practice questions

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Question 1Mathematical Processes and Number Sense

Which of the following numbers is irrational?

  1. √49
  2. √11 Correct answer.
  3. 0.75
  4. 22/7

Explanation

An irrational number cannot be written as a ratio of two integers, and its decimal expansion neither terminates nor repeats. The square root of 11 is not a perfect square, so it cannot be expressed as a fraction of integers and its decimal form is non-terminating and non-repeating. The other three values are all rational.

Question 2Mathematical Processes and Number Sense

A student claims that when you multiply any integer by 1, the product is that same integer. Which property of real numbers does this illustrate?

  1. Distributive property
  2. Multiplicative inverse
  3. Multiplicative identity Correct answer.
  4. Additive identity

Explanation

The multiplicative identity property states that multiplying any number by 1 leaves the number unchanged, since 1 is the identity element for multiplication. The additive identity involves adding 0, and the multiplicative inverse involves multiplying by a reciprocal to get 1.

Question 3Mathematical Processes and Number Sense

Evaluate: |−7| + |3| − |−2|.

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

Explanation

Absolute value gives the distance of a number from zero, so |−7| = 7, |3| = 3, and |−2| = 2. Substituting yields 7 + 3 − 2 = 8.

Question 4Mathematical Processes and Number Sense

Which list orders the numbers from least to greatest?

  1. −0.5, −3/4, 0.2, 5/8
  2. −3/4, −0.5, 0.2, 5/8 Correct answer.
  3. 0.2, 5/8, −0.5, −3/4
  4. −3/4, 0.2, −0.5, 5/8

Explanation

Converting to decimals gives −0.75, −0.5, 0.2, and 0.625. On a number line, −0.75 is farthest left, followed by −0.5, then 0.2, then 0.625. Thus the order from least to greatest is −3/4, −0.5, 0.2, 5/8.

Question 5Mathematical Processes and Number Sense

What is the prime factorization of 84?

  1. 2 × 3² × 7
  2. 2² × 21
  3. 2² × 3 × 7 Correct answer.
  4. 4 × 3 × 7

Explanation

Dividing 84 by 2 gives 42, dividing by 2 again gives 21, and 21 = 3 × 7. All factors 2, 3, and 7 are prime, so 84 = 2² × 3 × 7.

Question 6Mathematical Processes and Number Sense

Which of the following is the greatest common factor of 36 and 60?

  1. 180
  2. 18
  3. 6
  4. 12 Correct answer.

Explanation

The prime factorizations are 36 = 2² × 3² and 60 = 2² × 3 × 5. The GCF uses the lowest power of each shared prime: 2² × 3 = 12.

Question 7Mathematical Processes and Number Sense

A recipe uses 3/4 cup of sugar for every 2 cups of flour. What is the ratio of sugar to flour in simplest form?

  1. 2:3
  2. 3:8 Correct answer.
  3. 3:2
  4. 8:3

Explanation

The ratio of sugar to flour is (3/4) to 2, which can be written as the fraction (3/4)/2. Dividing gives 3/8, so the ratio is 3:8.

Question 8Mathematical Processes and Number Sense

Which expression correctly represents the statement 'five less than three times a number n'?

  1. 3n − 5 Correct answer.
  2. 5n − 3
  3. 3(n − 5)
  4. 5 − 3n

Explanation

'Three times a number n' is 3n, and 'five less than' that quantity means subtracting 5 from it. The correct translation is 3n − 5.

Question 9Mathematical Processes and Number Sense

Compute 2/3 ÷ 4/9.

  1. 3/2 Correct answer.
  2. 6/13
  3. 27/8
  4. 8/27

Explanation

Dividing by a fraction is equivalent to multiplying by its reciprocal, so 2/3 ÷ 4/9 = 2/3 × 9/4. Multiplying numerators and denominators gives 18/12, which simplifies to 3/2.

Question 10Mathematical Processes and Number Sense

A number is increased by 25% and the result is 90. What was the original number?

  1. 67.5
  2. 65
  3. 72 Correct answer.
  4. 112.5

Explanation

Increasing a number by 25% means multiplying it by 1.25. Solving 1.25x = 90 gives x = 90 ÷ 1.25 = 72.

Question 11Mathematical Processes and Number Sense

Using the order of operations, evaluate 18 − 3 × (4 − 6)² ÷ 6.

  1. 20
  2. 12
  3. 16 Correct answer.
  4. 15

Explanation

First evaluate the parentheses: 4 − 6 = −2. Squaring gives 4, so the expression becomes 18 − 3 × 4 ÷ 6. Multiplication and division proceed left to right: 3 × 4 = 12, then 12 ÷ 6 = 2, so 18 − 2 = 16.

Question 12Mathematical Processes and Number Sense

A shirt originally priced at $40 is on sale for 30% off. An additional 10% is taken off the sale price at the register. What is the final price?

  1. $24.00
  2. $28.00
  3. $26.00
  4. $25.20 Correct answer.

Explanation

After the 30% discount, the price is 40 × 0.70 = $28. The additional 10% is applied to $28, giving 28 × 0.90 = $25.20. Successive percent discounts multiply rather than add.

Question 13Mathematical Processes and Number Sense

Which of the following numbers is equivalent to 0.00045 written in scientific notation?

  1. 4.5 × 10³
  2. 4.5 × 10⁻⁴ Correct answer.
  3. 45 × 10⁻⁵
  4. 4.5 × 10⁻³

Explanation

Scientific notation requires a coefficient between 1 and 10 multiplied by a power of 10. Moving the decimal point in 0.00045 four places to the right gives 4.5, so the exponent is −4: 4.5 × 10⁻⁴.

Question 14Mathematical Processes and Number Sense

A store sells pencils in packs of 6 and erasers in packs of 8. What is the least number of pencils and erasers a teacher can buy so that each pencil is paired with exactly one eraser with none left over?

  1. 14
  2. 72
  3. 48
  4. 24 Correct answer.

Explanation

The number of each item must be a common multiple of 6 and 8, and the least such number is the least common multiple. The LCM of 6 = 2 × 3 and 8 = 2³ is 2³ × 3 = 24.

Question 15Mathematical Processes and Number Sense

Simplify the expression 3(2x − 4) − 2(x − 5).

  1. 8x − 2
  2. 4x − 22
  3. 4x + 2
  4. 4x − 2 Correct answer.

Explanation

Distributing gives 6x − 12 − 2x + 10. Combining like terms yields (6x − 2x) + (−12 + 10) = 4x − 2.

Question 16Mathematical Processes and Number Sense

A car travels 150 miles in 2.5 hours. At the same average rate, how far will it travel in 4 hours?

  1. 375 miles
  2. 225 miles
  3. 240 miles Correct answer.
  4. 300 miles

Explanation

The unit rate is 150 ÷ 2.5 = 60 miles per hour. Multiplying by 4 hours gives 60 × 4 = 240 miles.

Question 17Mathematical Processes and Number Sense

Which statement about the set of rational numbers is true?

  1. The square root of every rational number is rational.
  2. Rational numbers cannot be negative.
  3. Every integer is a rational number. Correct answer.
  4. Every rational number is an integer.

Explanation

Any integer n can be written as n/1, a ratio of two integers, so every integer is rational. The converse fails because fractions such as 1/2 are rational but not integers, and square roots of many rationals are irrational.

Question 18Mathematical Processes and Number Sense

A student writes 0.333... (repeating) and claims it is irrational because the decimal never ends. Which response is mathematically correct?

  1. The number is rational only if rounded to 0.33.
  2. The number is rational because it equals 1/3, a ratio of integers. Correct answer.
  3. The number is neither rational nor irrational.
  4. The student is right because non-terminating decimals are always irrational.

Explanation

A repeating decimal represents a rational number because it can be expressed as a fraction of integers. The repeating decimal 0.333... equals 1/3, so it is rational despite being non-terminating.

Question 19Mathematical Processes and Number Sense

What is 5.6 × 10³ written in standard decimal form?

  1. 0.0056
  2. 560
  3. 5,600 Correct answer.
  4. 56,000

Explanation

Multiplying by 10³ moves the decimal point three places to the right. Moving the decimal in 5.6 three places gives 5,600.

Question 20Mathematical Processes and Number Sense

A number line has tick marks equally spaced. Point P is located at −2 1/2 and point Q is located at 1 3/4. What is the distance between P and Q?

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

Explanation

Distance on a number line is the absolute value of the difference: |1 3/4 − (−2 1/2)|. Converting to quarters gives 1 3/4 = 7/4 and −2 1/2 = −10/4, so the difference is 17/4 = 4 1/4.

Question 21Mathematical Processes and Number Sense

Which of the following expressions has the greatest value?

  1. 5³ Correct answer.
  2. 2⁵
  3. 4³
  4. 3⁴

Explanation

Evaluating the powers gives 5³ = 125, 2⁵ = 32, 4³ = 64, and 3⁴ = 81. Comparing the numerical results shows that 125, from 5³, is greatest.

Question 22Mathematical Processes and Number Sense

A student needs to estimate √50 to the nearest whole number. Which reasoning correctly justifies the estimate?

  1. √50 is between 6 and 7, and since 50 is closer to 49 than to 36, the estimate is 7.
  2. √50 is between 7 and 8, and since 50 is closer to 49 than to 64, the estimate is 7. Correct answer.
  3. √50 is between 6 and 7, and since 50 is closer to 49 than to 64, the estimate is 7.
  4. √50 is between 7 and 8, and since 50 is closer to 64 than to 49, the estimate is 8.

Explanation

Since 7² = 49 and 8² = 64, √50 lies between 7 and 8. Because 50 is closer to 49 than to 64, √50 is closer to 7, so the nearest whole number is 7.

Question 23Mathematical Processes and Number Sense

Let a and b be nonzero real numbers. Which statement is always true?

  1. a ÷ b = b ÷ a
  2. a − b = b − a
  3. a + b = b + a Correct answer.
  4. (a + b) + c = a + (b + c) only when c = 0

Explanation

Addition of real numbers is commutative, so a + b always equals b + a. Subtraction and division are not commutative, and associativity of addition holds for all real c, not only c = 0.

Question 24Mathematical Processes and Number Sense

The repeating decimal 0.454545... can be written as a fraction in simplest form. What is that fraction?

  1. 9/22
  2. 45/100
  3. 5/11 Correct answer.
  4. 45/99

Explanation

Let x = 0.454545... Then 100x = 45.4545..., and subtracting gives 99x = 45, so x = 45/99. Reducing by the common factor 9 yields 5/11.

Question 25Mathematical Processes and Number Sense

A quantity is decreased by 20% and then the result is increased by 25%. Compared with the original quantity, the final quantity is which of the following?

  1. 5% less
  2. 1% greater
  3. 5% greater
  4. The same as the original Correct answer.

Explanation

A 20% decrease multiplies by 0.80, and a 25% increase multiplies by 1.25. The combined factor is 0.80 × 1.25 = 1.00, so the final quantity equals the original.

Question 26Mathematical Processes and Number Sense

Which set of numbers is closed under subtraction?

  1. The integers Correct answer.
  2. The whole numbers
  3. The natural numbers
  4. The positive rational numbers

Explanation

A set is closed under an operation when applying that operation to any two members always produces a member of the same set. Subtracting one integer from another always yields an integer, but each listed alternative set contains pairs whose difference falls outside it.

Question 27Mathematical Processes and Number Sense

What is the value of the expression |−7| + |4|?

  1. 11 Correct answer.
  2. 3
  3. −3
  4. −11

Explanation

Absolute value gives the distance of a number from zero, so |−7| = 7 and |4| = 4. Adding these distances gives 7 + 4 = 11.

Question 28Mathematical Processes and Number Sense

Which number is the additive inverse of −6?

  1. 1/6
  2. 0
  3. −6
  4. 6 Correct answer.

Explanation

The additive inverse of a number is the value that sums with it to zero. Since −6 + 6 = 0, the additive inverse is 6.

Question 29Mathematical Processes and Number Sense

Which of the following correctly applies the distributive property to 6(x + 4)?

  1. 10x
  2. 6x + 24 Correct answer.
  3. x + 24
  4. 6x + 4

Explanation

The distributive property multiplies the factor outside the parentheses by each term inside. So 6(x + 4) = 6·x + 6·4 = 6x + 24.

Question 30Mathematical Processes and Number Sense

Which of the following is the prime factorization of 84?

  1. 2 · 3² · 7
  2. 2² · 21
  3. 4 · 21
  4. 2² · 3 · 7 Correct answer.

Explanation

Repeatedly dividing 84 by its smallest prime factors gives 84 = 2 · 42 = 2 · 2 · 21 = 2² · 3 · 7. A prime factorization must use only prime factors.

Question 31Mathematical Processes and Number Sense

A number is multiplied by 10⁴. Which statement describes the effect on the decimal point?

  1. It moves 10 places to the right.
  2. It moves 4 places to the left.
  3. It does not move.
  4. It moves 4 places to the right. Correct answer.

Explanation

Multiplying by a power of 10 shifts the decimal point one place to the right for each factor of 10. Multiplying by 10⁴ therefore moves the decimal point 4 places to the right, making the number larger.

Question 32Mathematical Processes and Number Sense

Which inequality correctly compares −5/8 and −3/4?

  1. −5/8 ≥ −3/4
  2. −5/8 = −3/4
  3. −5/8 > −3/4 Correct answer.
  4. −5/8 < −3/4

Explanation

Writing both with a common denominator: −5/8 = −5/8 and −3/4 = −6/8. On the number line −5/8 lies to the right of −6/8, so −5/8 is greater.

Question 33Mathematical Processes and Number Sense

Which of the following is equivalent to 0.6%?

  1. 0.0006
  2. 0.6
  3. 0.006 Correct answer.
  4. 0.06

Explanation

To convert a percent to a decimal, divide by 100. So 0.6% = 0.6 ÷ 100 = 0.006.

Question 34Mathematical Processes and Number Sense

A class has 12 girls and 18 boys. What is the ratio of girls to the total number of students, in simplest form?

  1. 3:5
  2. 3:2
  3. 2:3
  4. 2:5 Correct answer.

Explanation

The total number of students is 12 + 18 = 30. The ratio of girls to total students is 12:30, which simplifies by dividing both terms by 6 to 2:5.

Question 35Mathematical Processes and Number Sense

A machine produces 45 parts in 6 minutes. At this constant rate, how many parts will it produce in 14 minutes?

  1. 90
  2. 105 Correct answer.
  3. 120
  4. 75

Explanation

The unit rate is 45 ÷ 6 = 7.5 parts per minute. Multiplying by 14 minutes gives 7.5 × 14 = 105 parts.

Question 36Mathematical Processes and Number Sense

An item costs $64 after a 20% discount. What was the price before the discount?

  1. $76.80
  2. $51.20
  3. $80.00 Correct answer.
  4. $84.00

Explanation

After a 20% discount the buyer pays 80% of the original price. Solving 0.80p = 64 gives p = 64 ÷ 0.80 = 80 dollars.

Question 37Mathematical Processes and Number Sense

Which expression represents the product of a number and its multiplicative inverse?

  1. x ÷ x²
  2. x + 1/x
  3. x · (1/x) Correct answer.
  4. x − 1/x

Explanation

The multiplicative inverse of a nonzero number x is 1/x, and their product is x · (1/x) = 1. This illustrates the inverse property of multiplication.

Question 38Mathematical Processes and Number Sense

Which expression is equivalent to 2⁵ · 2⁻³?

  1. 2² Correct answer.
  2. 4⁻²
  3. 2⁸
  4. 2⁻¹⁵

Explanation

When multiplying powers with the same base, add the exponents: 2⁵ · 2⁻³ = 2^(5 + (−3)) = 2².

Question 39Mathematical Processes and Number Sense

A number line is marked in equal increments of 1/4. Point A is at 1 and point B is at 2 3/4. How many increments apart are A and B?

  1. 7 Correct answer.
  2. 11
  3. 12
  4. 9

Explanation

The distance between the points is 2 3/4 − 1 = 1 3/4 = 7/4. Dividing by the increment 1/4 gives (7/4) ÷ (1/4) = 7 increments.

Question 40Mathematical Processes and Number Sense

A student claims that 0.5 is greater than 1/3 because 5 is greater than 3. Which reasoning identifies the error?

  1. Decimals are always greater than fractions.
  2. The fraction 1/3 should first be rounded to 1/2.
  3. The comparison should be made only with whole numbers.
  4. The values must be compared as quantities, not by the digits or terms used to write them. Correct answer.

Explanation

Comparing numerals digit by digit is invalid because 0.5 means one half and 1/3 means one third of the same whole. Since 1/2 > 1/3, the correct conclusion is that 0.5 is greater, but for the right reason.

Question 41Mathematical Processes and Number Sense

Which property justifies the statement 7 + (3 + 9) = (7 + 3) + 9?

  1. Identity property of addition
  2. Associative property of addition Correct answer.
  3. Commutative property of addition
  4. Distributive property

Explanation

The associative property of addition states that changing the grouping of addends does not change the sum. The order of the addends is unchanged here; only the parentheses move.

Question 42Mathematical Processes and Number Sense

A student estimates √75 by reasoning that 8² = 64 and 9² = 81. Which conclusion is best supported?

  1. √75 is exactly 8.5.
  2. √75 is greater than 9.
  3. √75 is between 8 and 9 and closer to 8.
  4. √75 is between 8 and 9 and closer to 9. Correct answer.

Explanation

Since 64 < 75 < 81, the square root lies between 8 and 9. Because 75 is closer to 81 than to 64, √75 is closer to 9.

Question 43Mathematical Processes and Number Sense

Which of the following is the prime factorization of 360?

  1. 2⁴ · 3 · 5
  2. 2² · 3² · 10
  3. 2³ · 45
  4. 2³ · 3² · 5 Correct answer.

Explanation

Dividing 360 by primes gives 360 = 8 · 9 · 5 = 2³ · 3² · 5, and every factor listed is prime. Checking the product confirms 2³ · 3² · 5 = 360.

Question 44Mathematical Processes and Number Sense

If x is a negative rational number and y is a positive rational number, which statement must be true?

  1. x · y is positive.
  2. x − y is positive.
  3. x ÷ y is positive.
  4. x · y is negative. Correct answer.

Explanation

The product of a negative number and a positive number is always negative, so x · y must be negative. The signs of the other expressions cannot be guaranteed or are determined incorrectly.

Question 45Mathematical Processes and Number Sense

The number 0.000305 is written in scientific notation as a × 10ⁿ. What are the values of a and n?

  1. a = 30.5, n = −5
  2. a = 0.305, n = −3
  3. a = 3.05, n = 4
  4. a = 3.05, n = −4 Correct answer.

Explanation

Scientific notation requires 1 ≤ |a| < 10, so the decimal point moves 4 places right to give 3.05. Moving right makes the exponent negative, so n = −4.

Question 46Mathematical Processes and Number Sense

A quantity is first increased by 50% and then the result is decreased by 50%. Compared with the original quantity, the final quantity is which of the following?

  1. 25% less Correct answer.
  2. The same
  3. 50% less
  4. 25% greater

Explanation

Let the original quantity be Q. After a 50% increase the value is 1.5Q, and after a 50% decrease it is 0.5(1.5Q) = 0.75Q. The final value is 75% of the original, which is 25% less.

Question 47Mathematical Processes and Number Sense

Which statement about the sum of a rational number and an irrational number is always true?

  1. The sum may be either rational or irrational.
  2. The sum is rational.
  3. The sum is zero.
  4. The sum is irrational. Correct answer.

Explanation

If the sum of a rational number r and an irrational number s were rational, then s = (sum) − r would be a difference of rationals and therefore rational, a contradiction. Hence the sum must be irrational.

Question 48Mathematical Processes and Number Sense

A student must place √2, 1.4, and 7/5 in order from least to greatest. Which ordering is correct?

  1. √2, 1.4, 7/5
  2. 1.4, 7/5, √2
  3. 1.4, √2, 7/5 Correct answer.
  4. 7/5, 1.4, √2

Explanation

Squaring the candidates: 1.4² = 1.96, (7/5)² = 1.96 as well, and 2 is greater than 1.96, so 1.4 = 7/5 < √2. Comparing 1.4 and 7/5 shows they are equal, so the strict order requires distinguishing them by another means.

Question 49Mathematical Processes and Number Sense

A number n is multiplied by 0.25. Which single operation produces the same result?

  1. Subtracting 0.75n from n
  2. Dividing n by 4 Correct answer.
  3. Dividing n by 0.25
  4. Multiplying n by 4

Explanation

Since 0.25 = 1/4, multiplying by 0.25 is the same as dividing by 4. Both operations take one quarter of the original quantity.

Question 50Mathematical Processes and Number Sense

A store sells notebooks in packs of 6 and pens in packs of 9. A teacher wants to buy the same number of notebooks and pens, using whole packs of each. What is the smallest total number of notebooks the teacher can buy?

  1. 3
  2. 18 Correct answer.
  3. 36
  4. 54

Explanation

The smallest number of notebooks equal to a whole number of pen packs is the least common multiple of 6 and 9. Since 6 = 2·3 and 9 = 3², the LCM is 2·3² = 18, so 3 notebook packs match 2 pen packs.