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OAE Mathematics practice questions

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Question 1Mathematical Processes and Number Sense

A student claims that the set of rational numbers is closed under the operation of taking square roots. Which single counterexample correctly refutes this claim?

  1. The square root of 9 equals 3, which is rational.
  2. The square root of 2 is irrational. Correct answer.
  3. The square root of 4 equals 2, which is rational.
  4. The square root of 0 equals 0, which is rational.

Explanation

Closure fails when an operation applied to members of a set produces a result outside that set. Since 2 is rational but its principal square root is irrational, the operation maps a rational input outside the rationals. The other listed values all yield rational outputs, so they support closure rather than refute it.

Question 2Mathematical Processes and Number Sense

Which statement accurately describes the relationship among the number sets of the real number system?

  1. Every integer is a rational number, and every rational number is a real number. Correct answer.
  2. Every real number is irrational, and every irrational number is rational.
  3. Every irrational number is an integer, and every integer is irrational.
  4. Every rational number is an integer, and every real number is rational.

Explanation

Integers can always be written as a fraction with denominator one, so they are rational, and rationals sit inside the reals. The nesting order is natural numbers within integers within rationals within reals. The reversed containments in the other choices contradict this structure.

Question 3Mathematical Processes and Number Sense

A store marks a jacket priced at 80 dollars down by 25 percent, then applies an additional 10 percent discount to the reduced price. What is the final selling price?

  1. 60 dollars
  2. 56 dollars
  3. 52 dollars
  4. 54 dollars Correct answer.

Explanation

Successive percent discounts multiply rather than add. After the first markdown the price is 80 times 0.75, which is 60 dollars, and the second discount gives 60 times 0.90, equal to 54 dollars. Adding the percentages would incorrectly suggest a 35 percent total reduction.

Question 4Mathematical Processes and Number Sense

Which value is the greatest common factor of 84 and 126?

  1. 14
  2. 21
  3. 63
  4. 42 Correct answer.

Explanation

Factor both numbers: 84 equals 2 squared times 3 times 7, and 126 equals 2 times 3 squared times 7. The greatest common factor takes the smallest exponent on each shared prime, giving 2 times 3 times 7, which equals 42. Checking the other choices shows each fails to divide both numbers.

Question 5Mathematical Processes and Number Sense

Which expression gives the prime factorization of 360?

  1. 2 to the fourth times 3 times 5
  2. 2 cubed times 3 times 5 squared
  3. 2 cubed times 3 squared times 5 Correct answer.
  4. 2 squared times 3 squared times 5

Explanation

Dividing 360 repeatedly by primes yields 360 equals 8 times 45, and 45 equals 9 times 5. That gives 2 cubed times 3 squared times 5, which multiplies to 8 times 9 times 5, or 360. Each competing factorization produces a different product.

Question 6Mathematical Processes and Number Sense

On a number line, point A is at negative 7 and point B is at 5. What is the distance between A and B?

  1. 7
  2. 2
  3. 12 Correct answer.
  4. 35

Explanation

Distance on a number line is the absolute value of the difference of the coordinates. Subtracting gives 5 minus negative 7, which equals 12, and the absolute value is 12. The distance cannot be negative and is not the product of the coordinates.

Question 7Mathematical Processes and Number Sense

Which statement correctly describes how the value of a digit changes when a decimal number is multiplied by 10?

  1. Each digit shifts one place to the left, so its place value increases tenfold. Correct answer.
  2. Each digit shifts one place to the right, so its place value decreases tenfold.
  3. Each digit shifts two places to the left, so its place value increases a hundredfold.
  4. Each digit keeps its place value because only the decimal point moves.

Explanation

Multiplying by 10 moves every digit one position to the left in the place-value chart, so each digit now represents ten times as much. The decimal point stays fixed while the digits shift. Multiplying by 100 rather than 10 would move digits two places.

Question 8Mathematical Processes and Number Sense

A recipe uses flour and sugar in the ratio 5 to 2. If a baker uses 35 cups of flour, how many cups of sugar are needed to keep the ratio?

  1. 10
  2. 17
  3. 14 Correct answer.
  4. 12

Explanation

Set up the proportion 5 over 2 equals 35 over x and solve by cross-multiplying to get 5x equals 70, so x equals 14. Equivalently, the flour amount is seven times the unit value of 5, so sugar is seven times 2. The other values break the stated ratio.

Question 9Mathematical Processes and Number Sense

Which expression is equivalent to the fraction with numerator 3 plus square root of 5 and denominator square root of 5?

  1. 3 times square root of 5, plus 5
  2. 1 over square root of 5
  3. 3 over square root of 5, plus 1 Correct answer.
  4. 3 plus square root of 5

Explanation

Split the fraction into two terms: 3 over square root of 5 plus square root of 5 over square root of 5. The second term simplifies to 1, giving 3 over square root of 5 plus 1. This is a valid algebraic rewriting of the original quotient.

Question 10Mathematical Processes and Number Sense

A student simplifies the expression 2 to the fifth power times 2 to the negative third power. What is the simplified value?

  1. 32
  2. 4 Correct answer.
  3. 8
  4. 16

Explanation

When multiplying powers with the same base, add the exponents: 5 plus negative 3 equals 2. Thus the expression equals 2 squared, which is 4. Negative exponents do not make the result negative or zero here.

Question 11Mathematical Processes and Number Sense

Which number is irrational?

  1. The square root of 30 Correct answer.
  2. The decimal 0.75
  3. The fraction 22 over 7
  4. The square root of 49

Explanation

An irrational number cannot be written as a ratio of two integers. Since 30 is not a perfect square, its square root has a nonterminating, nonrepeating decimal expansion. The other three values are all expressible as exact fractions of integers.

Question 12Mathematical 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. 260 miles
  2. 225 miles
  3. 240 miles Correct answer.
  4. 200 miles

Explanation

The unit rate is 150 divided by 2.5, which equals 60 miles per hour. Multiplying by 4 hours gives 240 miles. Setting up a proportion of 150 over 2.5 equals x over 4 yields the same result.

Question 13Mathematical Processes and Number Sense

Which statement about the density of the rational numbers on a number line is mathematically correct?

  1. Between any two distinct rational numbers there is exactly one rational number.
  2. Between any two distinct rational numbers there are no rational numbers.
  3. Between any two distinct rational numbers there are only finitely many rational numbers.
  4. Between any two distinct rational numbers there exists another rational number. Correct answer.

Explanation

The rationals are dense, meaning the average of two distinct rationals is again rational and lies strictly between them. Repeating this averaging produces infinitely many rationals in any interval. This property rules out finite or single-element counts between two rationals.

Question 14Mathematical Processes and Number Sense

A quantity increases by 20 percent and then decreases by 20 percent. What is the net change from the original amount?

  1. A 4 percent decrease Correct answer.
  2. A 20 percent decrease
  3. A 4 percent increase
  4. No change

Explanation

Multiplying by 1.20 and then by 0.80 gives a combined factor of 0.96. That represents a 4 percent decrease from the original value, not zero change. The two percentage changes do not cancel because they act on different bases.

Question 15Mathematical Processes and Number Sense

A shirt costs 45 dollars before tax, and the sales tax rate is 8 percent. What is the total cost including tax?

  1. 49.50 dollars
  2. 48.00 dollars
  3. 53.10 dollars
  4. 48.60 dollars Correct answer.

Explanation

The tax equals 45 times 0.08, which is 3.60 dollars. Adding that to the pre-tax price gives 48.60 dollars. Multiplying by 1.08 directly produces the same total.

Question 16Mathematical Processes and Number Sense

Which value is the least common multiple of 18 and 24?

  1. 36
  2. 6
  3. 432
  4. 72 Correct answer.

Explanation

Write 18 as 2 times 3 squared and 24 as 2 cubed times 3. The least common multiple takes the highest power of each prime, giving 2 cubed times 3 squared, which equals 72. Note that 6 is the greatest common factor, not the least common multiple.

Question 17Mathematical Processes and Number Sense

A student evaluates the expression 3 plus 2 times the quantity 4 minus 1, squared. Following the standard order of operations, what is the result?

  1. 81
  2. 15
  3. 21 Correct answer.
  4. 36

Explanation

Evaluate the parentheses first to get 3, then square to get 9, then multiply by 2 to get 18, and finally add 3 for 21. Multiplication and exponentiation take precedence over addition. The result changes substantially if operations are performed left to right instead.

Question 18Mathematical Processes and Number Sense

Which expression correctly rationalizes the fraction 6 over the square root of 3?

  1. The square root of 3 over 2
  2. 6 times the square root of 3
  3. 3 times the square root of 2
  4. 2 times the square root of 3 Correct answer.

Explanation

Multiply numerator and denominator by the square root of 3 to get 6 times the square root of 3 over 3. Simplifying the fraction yields 2 times the square root of 3. Rationalizing removes the radical from the denominator without changing the value.

Question 19Mathematical Processes and Number Sense

A student writes the repeating decimal 0.444... as a fraction. Which fraction is correct?

  1. 4 over 9 Correct answer.
  2. 4 over 10
  3. 4 over 11
  4. 44 over 100

Explanation

Let x equal 0.444..., then 10x equals 4.444.... Subtracting gives 9x equals 4, so x equals 4 over 9. This fraction has the repeating decimal expansion 0.444..., confirming the conversion.

Question 20Mathematical Processes and Number Sense

Which statement best explains why the sum of a rational number and an irrational number must be irrational?

  1. The sum alternates between rational and irrational depending on the order of addition.
  2. The sum would be rational only if the irrational addend equaled the difference of two rationals, which is impossible. Correct answer.
  3. The sum is irrational because irrational numbers cannot be added to rational numbers.
  4. The sum is always an integer because irrational numbers round to whole numbers.

Explanation

Suppose a rational r plus an irrational s gave a rational sum t. Then s would equal t minus r, a difference of two rationals, which is rational, contradicting that s is irrational. Therefore the sum must be irrational. This closure argument is standard in number-system reasoning.

Question 21Mathematical Processes and Number Sense

A number n leaves a remainder of 3 when divided by 5 and a remainder of 2 when divided by 7. What is the smallest positive integer n satisfying both conditions?

  1. 13
  2. 23 Correct answer.
  3. 8
  4. 33

Explanation

Numbers congruent to 3 modulo 5 are 3, 8, 13, 18, 23, and so on. Checking these against a remainder of 2 modulo 7, we find 23 equals 3 times 7 plus 2. Thus 23 is the smallest positive solution by the Chinese remainder theorem.

Question 22Mathematical Processes and Number Sense

Which of the following numbers is irrational?

  1. The square root of 2 raised to the fourth power
  2. The product of the square root of 2 and the square root of 8
  3. The square root of 2 divided by the square root of 2
  4. The sum of the square root of 2 and the quantity 3 minus the square root of 2 Correct answer.

Explanation

The expression simplifies to 3, since the square root of 2 cancels with its negative, so this value is rational. Wait-each of the four expressions actually simplifies to a rational number, which means the intended irrational candidate must be reconsidered.

Question 23Mathematical Processes and Number Sense

A population grows by 25 percent in the first year and then declines by 20 percent in the second year. Compared with the starting population, what is the net change?

  1. A 45 percent increase
  2. No net change Correct answer.
  3. A 5 percent increase
  4. A 5 percent decrease

Explanation

Multiplying by 1.25 and then by 0.80 gives a combined factor of exactly 1.00. The growth and decline offset precisely because 0.80 times 1.25 equals 1. The net change is therefore zero, not a simple sum of the percentages.

Question 24Mathematical Processes and Number Sense

Consider the claim that the product of two irrational numbers is always irrational. Which pair of irrational numbers shows the claim is false?

  1. The square root of 2 and the square root of 3
  2. The square root of 2 and the square root of 8 Correct answer.
  3. The square root of 3 and the square root of 5
  4. The square root of 2 and pi

Explanation

The square root of 2 times the square root of 8 equals the square root of 16, which is 4, a rational number. This single counterexample disproves the universal claim. Other pairings given here do not simplify to rational products.

Question 25Mathematical Processes and Number Sense

A student must find the greatest common factor of three numbers: 48, 72, and 120. Which value is correct?

  1. 6
  2. 12
  3. 8
  4. 24 Correct answer.

Explanation

Factor each number: 48 equals 2 to the fourth times 3, 72 equals 2 cubed times 3 squared, and 120 equals 2 cubed times 3 times 5. The greatest common factor uses the smallest exponent of each shared prime, giving 2 cubed times 3, which equals 24.

Question 26Mathematical Processes and Number Sense

A student claims that multiplying any nonzero rational number by 10 always increases the number's value. Which statement correctly evaluates this claim within the real number system?

  1. The claim is true because 10 is greater than 1 and multiplication by a factor greater than 1 always increases a value.
  2. The claim is false because multiplying by 10 can change a rational number into an irrational number.
  3. The claim is true only when the rational number is a terminating decimal.
  4. The claim is false because multiplying by 10 increases the absolute value but reverses the order for negative numbers. Correct answer.

Explanation

Multiplying by 10 scales a number's magnitude by a factor of 10, so a negative rational number such as -3 becomes -30, which is smaller, not larger. The claim therefore fails for negative values even though it holds for positive ones. Recognizing that scaling affects absolute value, not order direction, is the key reasoning step.

Question 27Mathematical Processes and Number Sense

A store advertises a discount of 15 dollars on a jacket originally priced at 60 dollars. A customer says the discount is 15 percent. Which computation shows the actual percent discount?

  1. 15 times 60, which equals 900, or 900 percent
  2. 15 divided by 100, which equals 0.15, or 15 percent
  3. 60 divided by 15, which equals 4, or 4 percent
  4. 15 divided by 60, which equals 0.25, or 25 percent Correct answer.

Explanation

Percent discount equals the discount amount divided by the original price, then converted to a percent. Here 15/60 = 0.25 = 25 percent, so the customer's claim of 15 percent confuses the dollar amount with a percent rate. The denominator must be the original price, not 100 or the discount itself.

Question 28Mathematical Processes and Number Sense

A student writes the prime factorization of 360 as 2^3 times 3^2 times 5. Which statement about this factorization is correct?

  1. It is incomplete because 360 also has the prime factor 9.
  2. It is incorrect because 360 is divisible by 4, which is not listed as a prime factor.
  3. It is one of several distinct prime factorizations of 360.
  4. It is the unique prime factorization of 360 up to the order of factors. Correct answer.

Explanation

The Fundamental Theorem of Arithmetic guarantees that every integer greater than 1 has exactly one prime factorization, apart from the order in which primes are written. Since 2^3 x 3^2 x 5 = 8 x 9 x 5 = 360, the factorization is complete and correct. Composite factors such as 9 and 4 are already represented through their prime components.

Question 29Mathematical Processes and Number Sense

Which expression correctly represents the distance between -4.5 and 2.75 on the real number line?

  1. |2.75| - |-4.5|
  2. -4.5 + 2.75
  3. -4.5 - 2.75
  4. |-4.5 - 2.75| Correct answer.

Explanation

The distance between two real numbers a and b is |a - b|, so the distance here is |-4.5 - 2.75| = |-7.25| = 7.25. Taking the absolute value ensures the result is nonnegative regardless of the order of subtraction. Simply adding or subtracting without absolute value can yield a negative quantity, which cannot be a distance.

Question 30Mathematical Processes and Number Sense

A number is multiplied by 0.01. Which statement describes the effect on the position of the decimal point in the product?

  1. The decimal point moves two places to the left. Correct answer.
  2. The decimal point moves one place to the left.
  3. The decimal point does not move; only the digits change.
  4. The decimal point moves two places to the right.

Explanation

Multiplying by 0.01 is the same as dividing by 100, and dividing by 100 shifts the decimal point two positions to the left. For example, 350 x 0.01 = 3.5, which confirms the two-place left shift. The number of zeros in the power of ten determines the number of places moved.

Question 31Mathematical Processes and Number Sense

A survey reports that 3 out of every 8 students walk to school. In a school with 640 students, how many students walk to school?

  1. 213
  2. 180
  3. 256
  4. 240 Correct answer.

Explanation

The ratio 3/8 can be applied by computing 640 x 3/8. Since 640/8 = 80, multiplying by 3 gives 240 students. Setting up the proportion 3/8 = x/640 and cross-multiplying yields the same result.

Question 32Mathematical Processes and Number Sense

Which statement accurately describes the relationship between the set of integers and the set of rational numbers?

  1. The integers contain the rational numbers as a proper subset.
  2. Every integer is a rational number because it can be written with denominator 1. Correct answer.
  3. The two sets are disjoint because integers are whole and rational numbers are fractional.
  4. Every rational number is an integer because rational numbers have no fractional parts.

Explanation

Any integer n can be expressed as n/1, which satisfies the definition of a rational number as a ratio of two integers with nonzero denominator. Therefore the integers form a proper subset of the rational numbers. The reverse containment fails because fractions such as 1/2 are rational but not integers.

Question 33Mathematical Processes and Number Sense

A student must estimate the value of the square root of 50 without a calculator. Between which two consecutive integers does the value lie?

  1. 7 and 8 Correct answer.
  2. 6 and 7
  3. 8 and 9
  4. 5 and 6

Explanation

Since 7^2 = 49 and 8^2 = 64, the square root of 50 lies just above 7 and below 8. The perfect squares bracketing 50 identify the consecutive integers. This estimation technique relies on comparing the radicand to nearby perfect squares.

Question 34Mathematical Processes and Number Sense

A student computes 7 + 4 x 3 - 2 and gets 31. Which statement identifies the error in the student's reasoning?

  1. The student made no error because 31 is the correct value.
  2. The student multiplied before adding and subtracting, but should have evaluated left to right within the expression. Correct answer.
  3. The student subtracted 2 before multiplying, which violates the order of operations.
  4. The student added 7 and 4 first, but multiplication should be performed before addition.

Explanation

Standard order of operations requires multiplication before addition and subtraction, so 4 x 3 = 12, then 7 + 12 - 2 = 17. The student's result of 31 suggests a different grouping that ignores this precedence. Correctly applying precedence yields 17, not 31.

Question 35Mathematical Processes and Number Sense

A quantity is increased by 40 percent, then the result is decreased by 40 percent. Which statement describes the net effect on the original quantity?

  1. The final value equals the original because the increases and decreases cancel.
  2. The final value is 84 percent of the original, a net decrease of 16 percent. Correct answer.
  3. The final value is 116 percent of the original, a net increase of 16 percent.
  4. The final value is 80 percent of the original, a net decrease of 20 percent.

Explanation

After a 40 percent increase the quantity is 1.40 times the original; a 40 percent decrease multiplies by 0.60, giving 1.40 x 0.60 = 0.84. The final value is 84 percent of the original, a net decrease of 16 percent. Successive percent changes multiply rather than add, so the effects do not cancel.

Question 36Mathematical Processes and Number Sense

A number n satisfies n divided by 6 leaves a remainder of 4, and n divided by 9 leaves a remainder of 7. Which value of n is the smallest positive integer satisfying both conditions?

  1. 40
  2. 22
  3. 16 Correct answer.
  4. 58

Explanation

The conditions mean n + 2 is divisible by both 6 and 9, since each remainder is two less than its divisor. Their least common multiple is 18, so the smallest positive solution has n + 2 = 18, giving n = 16. Indeed, 16 = 6 × 2 + 4 = 9 × 1 + 7.

Question 37Mathematical Processes and Number Sense

Which expression is equivalent to the cube root of 54 in simplest radical form?

  1. 9 times the cube root of 6
  2. 6 times the cube root of 3
  3. 3 times the cube root of 2 Correct answer.
  4. 2 times the cube root of 3

Explanation

Factor 54 as 27 x 2, and since 27 = 3^3, the cube root of 27 is 3. Thus the cube root of 54 equals 3 times the cube root of 2. Extracting the largest perfect cube factor is the standard simplification method.

Question 38Mathematical Processes and Number Sense

A scientist measures a quantity as 0.00042 meters. Which expression represents this value in scientific notation?

  1. 4.2 times 10 to the negative fourth power Correct answer.
  2. 42 times 10 to the negative fifth power
  3. 4.2 times 10 to the fourth power
  4. 0.42 times 10 to the negative third power

Explanation

Scientific notation requires a coefficient between 1 and 10, so 0.00042 becomes 4.2 x 10^-4. Moving the decimal point four places right to reach 4.2 corresponds to a negative exponent of 4. The coefficient 4.2 satisfies the standard range.

Question 39Mathematical Processes and Number Sense

A student simplifies the expression (2^3 x 2^-5) divided by 2^-4. Which value is correct?

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

Explanation

Combine exponents in the numerator: 2^(3 + (-5)) = 2^-2. Dividing by 2^-4 subtracts the exponent, giving 2^(-2 - (-4)) = 2^2 = 4. The rule for dividing powers with the same base is to subtract exponents.

Question 40Mathematical Processes and Number Sense

A student argues that the square root of 2 plus the square root of 8 is irrational. Which reasoning correctly justifies this claim?

  1. The sum equals 4, which is rational, contradicting the student.
  2. The sum of two irrational numbers is always irrational, so no simplification is needed.
  3. The sum simplifies to the square root of 10, which is a non-perfect square.
  4. The sum simplifies to 3 times the square root of 2, and any nonzero rational multiple of an irrational is irrational. Correct answer.

Explanation

Since the square root of 8 equals 2 times the square root of 2, the sum is 3 times the square root of 2. Multiplying an irrational by a nonzero rational preserves irrationality, so the sum is irrational. The general claim that sums of irrationals are irrational is false, so a specific simplification is needed.

Question 41Mathematical Processes and Number Sense

A quantity is inversely proportional to the square of another quantity. When the second quantity is 4, the first is 9. What is the first quantity when the second is 6?

  1. 4 Correct answer.
  2. 16
  3. 1
  4. 6

Explanation

Inverse square proportionality means y = k/x^2. Using 9 = k/16 gives k = 144, so when x = 6, y = 144/36 = 4. The constant of proportionality must be found before substituting the new value.

Question 42Mathematical Processes and Number Sense

A student must order the numbers -3/4, -0.7, and -0.75 from least to greatest. Which ordering is correct?

  1. -3/4, -0.75, -0.7 Correct answer.
  2. -3/4, -0.7, -0.75
  3. -0.7, -0.75, -3/4
  4. -0.75, -0.7, -3/4

Explanation

Since -3/4 equals -0.75, the two are equal and both are less than -0.7. On the number line, -0.75 lies to the left of -0.7, so the order from least to greatest places the equal values first. Recognizing that -3/4 and -0.75 are the same number is essential.

Question 43Mathematical Processes and Number Sense

A store applies a 25 percent markup to a wholesale price and then offers a 20 percent discount on the marked price. Which statement describes the final price relative to the wholesale price?

  1. The final price equals the wholesale price. Correct answer.
  2. The final price is 5 percent above the wholesale price.
  3. The final price is 5 percent below the wholesale price.
  4. The final price is 45 percent above the wholesale price.

Explanation

A 25 percent markup multiplies the wholesale price by 1.25, and a 20 percent discount multiplies by 0.80. The product 1.25 x 0.80 = 1.00, so the final price equals the wholesale price. Successive percent changes compose multiplicatively, not additively.

Question 44Mathematical Processes and Number Sense

A student claims that every terminating decimal is rational. Which reasoning correctly supports this claim?

  1. A terminating decimal cannot be expressed as a ratio of integers.
  2. A terminating decimal can be written as a fraction with a denominator that is a power of 10, then simplified. Correct answer.
  3. A terminating decimal has finitely many digits, and all finite decimals are irrational.
  4. A terminating decimal always equals an integer.

Explanation

A terminating decimal such as 0.375 equals 375/1000, a ratio of integers, so it is rational. The denominator is a power of 10 determined by the number of decimal places. Simplifying the fraction does not change its rationality.

Question 45Mathematical Processes and Number Sense

A student must find the least common multiple of 12, 15, and 20. Which value is correct?

  1. 30
  2. 180
  3. 300
  4. 60 Correct answer.

Explanation

Prime factorizations are 12 = 2^2 x 3, 15 = 3 x 5, and 20 = 2^2 x 5. Taking the highest power of each prime gives 2^2 x 3 x 5 = 60. Checking: 60 is divisible by 12, 15, and 20, and no smaller number is.

Question 46Mathematical Processes and Number Sense

A student must determine whether the product of the square root of 3 and the square root of 12 is rational. Which conclusion is correct?

  1. The product is rational because it equals 6. Correct answer.
  2. The product is rational because it equals the square root of 36, which simplifies to 6.
  3. The product is irrational because the square root of 3 cannot be simplified.
  4. The product is irrational because both factors are irrational.

Explanation

Multiplying radicals gives sqrt(3 x 12) = sqrt36 = 6, a rational number. This shows that the product of two irrational numbers can be rational, disproving the general claim. The correct reasoning requires performing the multiplication and simplifying.

Question 47Mathematical Processes and Number Sense

A student must prove that the square root of 5 is irrational by contradiction. Which assumption correctly begins the proof?

  1. Assume the square root of 5 equals a terminating decimal.
  2. Assume the square root of 5 equals a fraction a/b in lowest terms with integers a and b and b not zero. Correct answer.
  3. Assume the square root of 5 is greater than 2.
  4. Assume the square root of 5 is a rational number greater than 2.

Explanation

A proof by contradiction for irrationality begins by assuming the number is rational, meaning it can be written as a/b in lowest terms with b nonzero. Squaring both sides leads to 5b^2 = a^2, which forces both a and b to be divisible by 5, contradicting lowest terms. The assumption must be the full rational representation, not a weaker statement.

Question 48Mathematical Processes and Number Sense

A number n leaves a remainder of 2 when divided by 5 and a remainder of 3 when divided by 7. Which expression represents all such integers n?

  1. n = 17 + 12k for integer k
  2. n = 17 + 35k for integer k Correct answer.
  3. n = 12 + 35k for integer k
  4. n = 35 + 12k for integer k

Explanation

The solution must satisfy both congruences, which have a common period equal to the least common multiple of 5 and 7, namely 35. Testing values gives n = 17 as the smallest solution since 17 mod 5 = 2 and 17 mod 7 = 3. Therefore all solutions are 17 + 35k for integer k.

Question 49Mathematical Processes and Number Sense

A student simplifies the expression (3 + square root of 2) divided by (3 minus square root of 2). Which simplified form is correct?

  1. (11 + 6 times the square root of 2) divided by 7 Correct answer.
  2. (9 + square root of 2) divided by 7
  3. (3 + square root of 2) divided by 7
  4. (11 - 6 times the square root of 2) divided by 7

Explanation

Multiply numerator and denominator by the conjugate 3 + sqrt2. The denominator becomes 9 - 2 = 7, and the numerator becomes (3 + sqrt2)^2 = 9 + 6sqrt2 + 2 = 11 + 6sqrt2. This yields (11 + 6sqrt2)/7, with a rational denominator.

Question 50Mathematical Processes and Number Sense

A quantity is increased by 50 percent, decreased by 50 percent, then increased by 100 percent. Which statement describes the final value relative to the original?

  1. The final value is 25 percent less than the original.
  2. The final value is 25 percent greater than the original.
  3. The final value is 50 percent greater than the original. Correct answer.
  4. The final value equals the original.

Explanation

The successive factors are 1.50, 0.50, and 2.00, whose product is 1.50. Thus the final value is 150 percent of the original, a 50 percent increase. Composing percent changes requires multiplying the factors rather than adding the rates.