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Mathematics Basic practice questions

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Question 1Number and Quantity

Which of the following numbers is classified as an irrational number?

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

Explanation

An irrational number cannot be written as a ratio of two integers, and its decimal expansion is non-terminating and non-repeating. The square root of 2 has such an expansion, so it is irrational. The other three values are all rational.

Question 2Number and Quantity

What is the value of |−7| + |4|?

  1. 28
  2. −3
  3. 11 Correct answer.
  4. 3

Explanation

Absolute value gives the distance of a number from zero, so |−7| = 7 and |4| = 4. Adding these gives 7 + 4 = 11. Absolute value is never negative, so the sum must be positive.

Question 3Number and Quantity

Which expression correctly writes 0.00045 in scientific notation?

  1. 45 × 10^−5
  2. 4.5 × 10^−4 Correct answer.
  3. 4.5 × 10^4
  4. 0.45 × 10^−3

Explanation

Scientific notation requires a coefficient between 1 and 10 multiplied by a power of ten. Moving the decimal point four places right to reach 4.5 means the exponent is −4. Thus 0.00045 = 4.5 × 10^−4.

Question 4Number and Quantity

What is the greatest common divisor of 36 and 60?

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

Explanation

Factoring gives 36 = 2² × 3² and 60 = 2² × 3 × 5. The GCD takes the lowest power of each shared prime, giving 2² × 3 = 12. Checking, 36 ÷ 12 = 3 and 60 ÷ 12 = 5, both integers.

Question 5Number and Quantity

A shirt originally priced at 80 dollars is discounted by 25 percent. What is the sale price in dollars?

  1. 65
  2. 20
  3. 60 Correct answer.
  4. 55

Explanation

A 25 percent discount removes 0.25 × 80 = 20 dollars from the original price. Subtracting gives 80 − 20 = 60 dollars. Equivalently, the buyer pays 75 percent of 80, which is also 60.

Question 6Number and Quantity

Which list orders the numbers −5, 3, 0, and −2 from least to greatest?

  1. 0, −2, 3, −5
  2. −5, −2, 0, 3 Correct answer.
  3. 3, 0, −2, −5
  4. −2, −5, 0, 3

Explanation

On the number line, values increase as you move right. The most negative value is −5, followed by −2, then 0, then the positive value 3. So the correct order is −5, −2, 0, 3.

Question 7Number and Quantity

What is the value of 2^5?

  1. 10
  2. 32 Correct answer.
  3. 25
  4. 64

Explanation

The expression 2^5 means multiplying five factors of 2: 2 × 2 × 2 × 2 × 2. That product equals 32. The exponent counts factors, it does not multiply the base by the exponent.

Question 8Number and Quantity

Which of these numbers is a prime number?

  1. 51
  2. 21
  3. 1
  4. 29 Correct answer.

Explanation

A prime number has exactly two distinct positive divisors: 1 and itself. The number 29 is not divisible by 2, 3, or 5, and √29 is less than 6, so it is prime. The other options each have additional divisors.

Question 9Number and Quantity

Compute the value of (−3) × (−4) + (−2).

  1. 10 Correct answer.
  2. 14
  3. −14
  4. −10

Explanation

Multiplication is performed before addition. The product of two negative numbers is positive, so (−3) × (−4) = 12. Then 12 + (−2) = 10.

Question 10Number and Quantity

Simplify the expression (2/3) ÷ (4/9).

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

Explanation

Dividing by a fraction is the same as multiplying by its reciprocal. So (2/3) ÷ (4/9) = (2/3) × (9/4) = 18/12, which simplifies to 3/2. The result is greater than 1 because the divisor is smaller than the dividend.

Question 11Number and Quantity

A recipe uses 3 cups of flour for every 4 cups of milk. If a cook uses 18 cups of flour, how many cups of milk are needed at the same ratio?

  1. 24 Correct answer.
  2. 20
  3. 22
  4. 13.5

Explanation

The ratio of flour to milk is 3:4. Scaling flour from 3 to 18 multiplies by 6, so milk must also be multiplied by 6, giving 4 × 6 = 24 cups. Checking, 18/24 simplifies to 3/4, confirming the ratio holds.

Question 12Number and Quantity

A quantity increases from 40 to 50. What is the percent increase?

  1. 80 percent
  2. 25 percent Correct answer.
  3. 20 percent
  4. 10 percent

Explanation

Percent change is the change divided by the original amount, times 100. The change is 50 − 40 = 10, and 10/40 = 0.25, which is a 25 percent increase. Dividing by the original value, not the new value, is essential.

Question 13Number and Quantity

Which expression is equivalent to (3 × 10^4) × (2 × 10^3)?

  1. 6 × 10^7 Correct answer.
  2. 6 × 10^8
  3. 6 × 10^12
  4. 5 × 10^7

Explanation

When multiplying numbers in scientific notation, multiply the coefficients and add the exponents. The coefficients give 3 × 2 = 6, and the exponents give 10^(4+3) = 10^7. The result 6 × 10^7 is already in proper form.

Question 14Number and Quantity

Simplify the expression 5 + 2 × (8 − 3)^2 using the standard order of operations.

  1. 175
  2. 35
  3. 105
  4. 55 Correct answer.

Explanation

Evaluate the parentheses first: 8 − 3 = 5. Then apply the exponent: 5² = 25. Multiply before adding: 2 × 25 = 50, and finally 5 + 50 = 55.

Question 15Number and Quantity

A number is multiplied by 4 and then 7 is subtracted, giving a result of 29. What is the original number?

  1. 9 Correct answer.
  2. 36
  3. 5.5
  4. 7

Explanation

Let x be the number, so 4x − 7 = 29. Adding 7 to both sides gives 4x = 36, and dividing by 4 gives x = 9. Checking, 4 × 9 − 7 = 36 − 7 = 29, which confirms the answer.

Question 16Number and Quantity

Which value of x satisfies the inequality 3x − 5 < 10?

  1. x < 5 Correct answer.
  2. x > 15
  3. x < 15
  4. x > 5

Explanation

Add 5 to both sides to get 3x < 15, then divide by the positive number 3, which preserves the inequality direction. This gives x < 5. Any value less than 5, such as 0, makes the original inequality true.

Question 17Number and Quantity

Simplify (x^3 · x^5) / x^2, assuming x is nonzero.

  1. x^10
  2. x^16
  3. x^6 Correct answer.
  4. x^4

Explanation

When multiplying powers with the same base, add the exponents: x^3 · x^5 = x^8. When dividing, subtract the exponents: x^8 / x^2 = x^(8−2) = x^6. The nonzero condition ensures no division by zero occurs.

Question 18Number and Quantity

What is the decimal expansion of the rational number 5/8?

  1. 0.58
  2. 0.625 Correct answer.
  3. 0.85
  4. 1.6

Explanation

Dividing 5 by 8 gives 0.625 exactly, so the decimal terminates. Because the denominator 8 = 2³ has only 2 as a prime factor, the decimal expansion must terminate within three places. This confirms the result.

Question 19Number and Quantity

The distance from Earth to the Sun is about 1.5 × 10^8 kilometers. How many kilometers is that in standard decimal form?

  1. 1,500,000,000
  2. 15,000
  3. 150,000,000 Correct answer.
  4. 15,000,000

Explanation

Multiplying 1.5 by 10^8 moves the decimal point eight places to the right. Starting from 1.5, that gives 150,000,000. The exponent 8 indicates eight zeros' worth of magnitude, not eight digits in the coefficient.

Question 20Number and Quantity

A store marks up the wholesale cost of an item by 40 percent, then later discounts the marked price by 20 percent. Compared with the original wholesale cost, the final price is:

  1. the same
  2. 12 percent higher Correct answer.
  3. 20 percent lower
  4. 20 percent higher

Explanation

Let the wholesale cost be 100. A 40 percent markup gives 140, and a 20 percent discount on 140 gives 140 × 0.8 = 112. The final price is 112 versus the original 100, which is 12 percent higher. Successive percent changes multiply rather than add.

Question 21Number and Quantity

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

  1. It is always rational.
  2. It is always an integer.
  3. It is always irrational. Correct answer.
  4. It may be either rational or irrational depending on the values.

Explanation

Suppose a nonzero rational r times an irrational i were rational, say q. Then i = q/r would be a ratio of rationals, hence rational, contradicting that i is irrational. So the product must be irrational whenever the rational factor is nonzero.

Question 22Number and Quantity

A number n leaves a remainder of 3 when divided by 5 and a remainder of 2 when divided by 7. Which of the following could be n?

  1. 23 Correct answer.
  2. 45
  3. 32
  4. 18

Explanation

Check 23: dividing by 5 leaves remainder 3, and dividing by 7 leaves remainder 2, so both conditions hold. The other candidates fail at least one remainder condition. Testing each candidate directly is a reliable strategy for such small numbers.

Question 23Number and Quantity

If a and b are both negative integers, which statement about a − b is always true?

  1. The result is always negative.
  2. The result is always positive.
  3. The sign of the result cannot be determined from the given information. Correct answer.
  4. The result is always zero.

Explanation

Subtracting a negative number is the same as adding its positive counterpart, so a − b = a + |b|. Depending on which magnitude is larger, the result can be positive, negative, or zero. For example, −2 − (−5) = 3 but −5 − (−2) = −3, so no single sign is guaranteed.

Question 24Number and Quantity

A quantity doubles every 3 hours. If it starts at 5 units, which expression gives the number of units after t hours?

  1. 5 · 2^(3t)
  2. 5 · 3^(t/2)
  3. 5 · 2^(t/3) Correct answer.
  4. 5 + 2t

Explanation

The doubling period is 3 hours, so the number of doubling intervals in t hours is t/3. Each interval multiplies the quantity by 2, giving 5 · 2^(t/3). Checking at t = 3 gives 5 · 2 = 10, consistent with one doubling.

Question 25Number and Quantity

The value of √50 lies between which two consecutive integers?

  1. 6 and 7
  2. 7 and 8 Correct answer.
  3. 24 and 26
  4. 8 and 9

Explanation

Since 7² = 49 and 8² = 64, and 50 lies between 49 and 64, the square root of 50 must lie between 7 and 8. In fact √50 ≈ 7.07, just above 7. Comparing squares of integers is the standard way to bound a square root.

Question 26Number and Quantity

Which expression correctly applies the distributive property to rewrite 6(2x + 5)?

  1. 12x + 30 Correct answer.
  2. 8x + 11
  3. 12x + 11
  4. 12x + 5

Explanation

The distributive property states that a(b + c) = ab + ac. Multiplying 6 by each term inside the parentheses gives 6·2x = 12x and 6·5 = 30, so the expression becomes 12x + 30.

Question 27Number and Quantity

Which number is the additive inverse of −7?

  1. 7 Correct answer.
  2. −7
  3. 0
  4. 1/7

Explanation

The additive inverse of a number is the value that, when added to the original number, gives zero. Since −7 + 7 = 0, the additive inverse of −7 is 7.

Question 28Number and Quantity

What is the value of |−9| − |4|?

  1. −13
  2. −5
  3. 13
  4. 5 Correct answer.

Explanation

The absolute value of a number is its distance from zero on the number line, so |−9| = 9 and |4| = 4. Subtracting gives 9 − 4 = 5.

Question 29Number and Quantity

A number is multiplied by 3, and then 4 is added. The result is 19. What is the number?

  1. 23
  2. 15
  3. 5 Correct answer.
  4. 7

Explanation

Let the number be x. The operations give the equation 3x + 4 = 19. Subtracting 4 from both sides gives 3x = 15, and dividing by 3 gives x = 5.

Question 30Number and Quantity

If x is a negative integer, which of the following must also be a negative integer?

  1. x^3 Correct answer.
  2. −x
  3. |x|
  4. x^2

Explanation

A negative number raised to an odd power remains negative, so x^3 is negative for any negative integer x. The other expressions all yield nonnegative values: x^2 and |x| are positive (or zero, though x is nonzero), and −x is positive.

Question 31Algebra

Simplify the expression 3(2x - 4) + 5x.

  1. 11x - 12 Correct answer.
  2. 6x - 12
  3. 11x + 12
  4. 11x - 4

Explanation

Distribute the 3 to get 6x - 12, then combine like terms with 5x to obtain 11x - 12. Only the x-terms combine; the constant -12 stays unchanged.

Question 32Algebra

What is the value of x in the linear equation 4x - 7 = 21?

  1. 14
  2. 28
  3. 7 Correct answer.
  4. 3.5

Explanation

Add 7 to both sides to get 4x = 28, then divide both sides by 4 to get x = 7. Each step must be applied to both sides to keep the equation balanced.

Question 33Algebra

Which expression represents the slope of the line passing through the points (2, 5) and (6, 13)?

  1. (6 + 2)/(13 + 5)
  2. (6 - 2)/(13 - 5)
  3. (13 + 5)/(6 + 2)
  4. (13 - 5)/(6 - 2) Correct answer.

Explanation

Slope is the change in y divided by the change in x, written as (y2 - y1)/(x2 - x1). Substituting the points gives (13 - 5)/(6 - 2), which equals 2.

Question 34Algebra

Evaluate the expression 2x^2 - 3y when x = 3 and y = 4.

  1. 18
  2. 42
  3. 30
  4. 6 Correct answer.

Explanation

Substitute the values to get 2(9) - 3(4), which is 18 - 12 = 6. The exponent applies to x before multiplying by 2.

Question 35Algebra

Which of the following is a factor of the quadratic expression x^2 - 5x + 6?

  1. x + 6
  2. x + 3
  3. x - 2 Correct answer.
  4. x - 6

Explanation

The expression factors as (x - 2)(x - 3) because the two numbers multiply to 6 and add to -5. Therefore x - 2 is a factor.

Question 36Algebra

Solve the inequality 3x + 4 < 19.

  1. x < 23/3
  2. x < 5 Correct answer.
  3. x > 5
  4. x > 23/3

Explanation

Subtract 4 from both sides to get 3x < 15, then divide by 3 to get x < 5. Dividing by a positive number does not flip the inequality sign.

Question 37Algebra

What is the y-intercept of the line given by the equation 2x + 3y = 12?

  1. (0, 4) Correct answer.
  2. (6, 0)
  3. (4, 0)
  4. (0, 6)

Explanation

The y-intercept occurs where x = 0. Substituting gives 3y = 12, so y = 4, and the point is (0, 4).

Question 38Algebra

Simplify the expression (x^3)(x^4).

  1. x
  2. x^7 Correct answer.
  3. 2x^7
  4. x^12

Explanation

When multiplying powers with the same base, add the exponents: 3 + 4 = 7, giving x^7. The base stays the same.

Question 39Algebra

Find the solution set of the equation x^2 - 9 = 0.

  1. {-3}
  2. {3, -3} Correct answer.
  3. {9, -9}
  4. {3}

Explanation

The equation factors as (x - 3)(x + 3) = 0, so x = 3 or x = -3. Both values satisfy the original equation.

Question 40Algebra

If f(x) = 2x + 1 and g(x) = x^2, what is f(g(3))?

  1. 19 Correct answer.
  2. 37
  3. 13
  4. 49

Explanation

First evaluate the inner function: g(3) = 9. Then apply f to that result: f(9) = 2(9) + 1 = 19.

Question 41Algebra

Solve the system of equations: x + y = 10 and x - y = 4.

  1. x = 6, y = 4
  2. x = 3, y = 7
  3. x = 5, y = 5
  4. x = 7, y = 3 Correct answer.

Explanation

Adding the two equations eliminates y, giving 2x = 14, so x = 7. Substituting into x + y = 10 gives y = 3.

Question 42Algebra

Which expression is equivalent to (2x - 3)^2?

  1. 4x^2 - 9
  2. 4x^2 - 6x + 9
  3. 4x^2 + 9
  4. 4x^2 - 12x + 9 Correct answer.

Explanation

Squaring the binomial gives (2x)^2 - 2(2x)(3) + 3^2, which simplifies to 4x^2 - 12x + 9. The middle term comes from doubling the product of the two terms.

Question 43Algebra

What is the slope of a line perpendicular to the line y = (2/3)x + 5?

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

Explanation

Perpendicular lines have slopes that are negative reciprocals. The negative reciprocal of 2/3 is -3/2.

Question 44Algebra

Factor the quadratic expression 2x^2 + 7x + 3.

  1. (2x + 7)(x + 3)
  2. (2x + 1)(x + 2)
  3. (2x + 1)(x + 3) Correct answer.
  4. (2x + 3)(x + 1)

Explanation

Expanding (2x + 1)(x + 3) gives 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3. The inner and outer products combine to the correct middle term.

Question 45Algebra

Solve for x: 5/(x - 1) = 10/4.

  1. x = 1.5
  2. x = 3 Correct answer.
  3. x = 5
  4. x = 2

Explanation

Cross-multiply to get 5(4) = 10(x - 1), so 20 = 10x - 10. Adding 10 gives 30 = 10x, so x = 3.

Question 46Algebra

The quadratic equation x^2 - 6x + 9 = 0 has which type of solutions?

  1. Two complex solutions
  2. No solutions at all
  3. Two distinct real solutions
  4. One repeated real solution Correct answer.

Explanation

The discriminant is b^2 - 4ac = 36 - 36 = 0, which means the quadratic has exactly one repeated real root. Here it factors as (x - 3)^2.

Question 47Algebra

If 3^(x+1) = 27, what is the value of x?

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

Explanation

Rewrite 27 as 3^3 so that both sides have base 3. Then x + 1 = 3, giving x = 2.

Question 48Algebra

What is the domain of the function f(x) = sqrt(x - 4)?

  1. x > 4
  2. x <= 4
  3. All real numbers
  4. x >= 4 Correct answer.

Explanation

The expression under a square root must be nonnegative, so x - 4 >= 0, which gives x >= 4. The value 4 is included because the square root of zero is defined.

Question 49Algebra

Simplify the rational expression (x^2 - 4)/(x - 2), assuming x is not equal to 2.

  1. 2x
  2. x + 2 Correct answer.
  3. x^2 + 2
  4. x - 2

Explanation

Factor the numerator as (x - 2)(x + 2), then cancel the common factor (x - 2). The result is x + 2 for x not equal to 2.

Question 50Algebra

A line passes through the point (3, -2) and has slope -4. Which equation represents this line in point-slope form?

  1. y + 2 = 4(x - 3)
  2. y + 2 = -4(x - 3) Correct answer.
  3. y - 3 = -4(x + 2)
  4. y - 2 = -4(x + 3)

Explanation

Point-slope form is y - y1 = m(x - x1). Substituting m = -4 and (x1, y1) = (3, -2) gives y + 2 = -4(x - 3), since subtracting -2 yields y + 2.