GED Mathematical Reasoning

Question 1

What is the value of 2/5 multiplied by ¾ divide by 8/5

Your Answer: Option(s)

Correct Answer: Option(s) C

Rationale

2/5 multiplied by ¾ divided by 8/5 equals 316.

To solve the expression, we first multiply 2/5 by 3/4, which results in 6/20 or 3/10. Then, we divide this result by 8/5, which is equivalent to multiplying by the reciprocal (5/8). This gives us (3/10) * (5/8) = 15/80, which simplifies to 3/16. The numeric value of 3/16 is approximately 0.1875, which equates to 316 when expressed in a decimal format for the given problem context.

A) 1225
This choice is incorrect as it does not accurately represent the outcome of the operations performed. The result of multiplying and dividing fractions yields a much smaller value than 1225, which is not possible given the fractions involved.

B) 13
Choosing 13 is incorrect because the calculations lead to a fractional result significantly less than 13. The multiplication and division of fractions consistently produce values between 0 and 1, and thus cannot reach the integer value of 13.

C) 316
This choice is indeed correct. The operations performed correctly lead to a value that aligns with the numerical representation of the expression. The resultant fraction, when calculated accurately, corresponds to this answer.

D) 64/75
This option is incorrect because the calculations do not yield a result equivalent to 64/75. The operations performed reduce the value far below this fraction, making it an inconsistency in the context of the problem.

Conclusion
The value of the expression 2/5 multiplied by ¾ divided by 8/5 simplifies through fraction multiplication and division, correctly yielding a final value of 316. Each incorrect option fails to align with the mathematical operations applied, reinforcing the accuracy of the calculations leading to the correct answer.

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Question 2

Simplify 6^2 - 3^2

Your Answer: Option(s)

Correct Answer: Option(s) C

Rationale

6^2 - 3^2 simplifies to 27.

Using the difference of squares formula, \( a^2 - b^2 = (a-b)(a+b) \), where \( a = 6 \) and \( b = 3 \), we find \( 6^2 - 3^2 = (6-3)(6+3) = 3 \times 9 = 27 \).

A) 6
This choice results from an incorrect interpretation of the difference of squares. It does not reflect the outcome of subtracting the squares of the numbers involved. It is essential to apply the correct algebraic formula to reach the right solution.

B) 9
Choosing 9 might suggest a misunderstanding of the operations involved. While \( 3^2 = 9 \), the question specifically asks for the difference between the squares of 6 and 3, not just the square of 3. The calculation must account for both squares to arrive at the correct answer.

C) 27
This choice correctly represents the result of simplifying \( 6^2 - 3^2 \). Using the difference of squares formula, we calculate \( 36 - 9 \) to achieve \( 27 \). Thus, this is the correct simplification.

D) 3
This option may arise from an incorrect subtraction of the two squares directly, interpreting \( 6 - 3 = 3 \). However, this approach neglects the squaring operation required in the question, leading to an inaccurate conclusion.

Conclusion
The expression \( 6^2 - 3^2 \) simplifies to 27 by applying the difference of squares formula correctly. The other choices reflect misunderstandings of either the operations or the algebraic principles involved. Mastery of these concepts is crucial for accurate simplification in mathematics.

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Question 3

2^3 * 27^(1/3) * 1^3

Your Answer: Option(s)

Correct Answer: Option(s) B

Rationale

2^3 * 27^(1/3) * 1^3 equals 24.

To solve the expression, we first evaluate each component: \(2^3 = 8\), \(27^{1/3} = 3\), and \(1^3 = 1\). Multiplying these results together gives \(8 * 3 * 1 = 24\).

A) 54
The choice of 54 may arise from incorrect calculations, such as mistakenly multiplying components or misinterpreting the cube root. The proper calculation yields 24, and 54 does not correspond to any correct combination of the components given in the expression.

B) 24
This is the correct answer, derived from \(2^3 = 8\), \(27^{1/3} = 3\), and \(1^3 = 1\). Multiplying these values together results in \(8 * 3 * 1 = 24\), confirming that this is the accurate evaluation of the expression.

C) 72
Choosing 72 suggests an error in multiplication or a misunderstanding of the cube root operation. The correct evaluation of the expression does not yield 72, as the factors involved do not combine to produce this value.

D) 18
The selection of 18 indicates a possible miscalculation, perhaps from incorrectly evaluating one of the components or misapplying the order of operations. The actual computed result of the expression is 24, not 18.

Conclusion
The expression \(2^3 * 27^{1/3} * 1^3\) evaluates clearly to 24 through proper calculation of each component. The correct result is a product of \(2^3\), the cube root of 27, and the cube of 1, all of which combine to yield 24. The other options reflect common errors in arithmetic or misunderstanding of the operations involved.

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Question 4

((5^3 * 2^4)^2)(5^(-2) * 2^5)

Your Answer: Option(s)

Correct Answer: Option(s) C

Rationale

Extracting the expression results in 5^4 * 2^13.

When simplifying the given expression ((5^3 * 2^4)^2)(5^(-2) * 2^5), we first apply the power of a product rule and then combine like bases to arrive at the final simplified form of 5^4 * 2^13.

A) 5^3 * 2^11
This choice results from an incorrect simplification of the expression. The calculation mistakenly does not account for the squaring of both bases in the first term or the full combination of exponents from both parts of the expression.

B) 5^(-12) * 2^40
This option represents an erroneous outcome due to miscalculating the exponents. The negative exponent for the base of 5 suggests a misunderstanding of the multiplication and simplification process, as the correct operations yield positive exponents.

C) 5^4 * 2^13
This is the correct choice, as it properly reflects the result of applying the rules of exponents. Simplifying the expression yields 5^(3*2 - 2) = 5^4 and 2^(4*2 + 5) = 2^13, confirming the accuracy of the calculation.

D) (-5)^8 * 2^13
This choice incorrectly incorporates a negative base in the power of 5. The original expression does not involve a negative component, so the transition to (-5)^8 is not justified and does not align with the operations performed.

Conclusion
Upon simplification of the expression ((5^3 * 2^4)^2)(5^(-2) * 2^5), we find that the correct outcome is 5^4 * 2^13. Understanding the laws of exponents is crucial in obtaining the correct answer, as demonstrated by the incorrect answers that arise from misapplication of these rules.

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Question 5

The mass of an amoeba is approximately 4.0 X 10^(-6) grams. Approximately how many amoebas are present in a sample that weighs 1 gram?

Your Answer: Option(s)

Correct Answer: Option(s) A

Rationale

Approximately 2.5 X 10^5 amoebas are present in a sample that weighs 1 gram.

To find the number of amoebas in a 1-gram sample, we divide the total mass of the sample (1 gram) by the mass of one amoeba (4.0 X 10^(-6) grams). This calculation yields approximately 2.5 X 10^5 amoebas.

A) 2.5 X 10^5
This choice correctly represents the number of amoebas calculated by dividing 1 gram by the mass of an individual amoeba (4.0 X 10^(-6) grams). Performing the calculation gives us 1 gram / (4.0 X 10^(-6) grams) = 2.5 X 10^5 amoebas.

B) 4.0 X 10^7
This answer suggests an incorrect calculation. If we mistakenly multiplied the mass of the amoeba instead of dividing, we would arrive at a much larger number, which does not accurately reflect the number of amoebas in the sample.

C) 4.0 X 10^5
While this choice is a plausible figure, it is based on an incorrect calculation. This result may reflect a misunderstanding of the division process needed to determine the number of amoebas from the total mass and the mass of a single amoeba.

D) 2.5 X 10^7
This option represents an exaggerated value and likely results from either multiplication errors or a misinterpretation of the scaling of the amoeba's mass. The correct approach clearly shows that the actual number of amoebas is significantly lower.

Conclusion
The correct calculation shows that approximately 2.5 X 10^5 amoebas can be found in a 1-gram sample based on their individual mass of 4.0 X 10^(-6) grams. Understanding the fundamental principle of division in this context is crucial for accurately determining the number of entities in a given mass. The other options reflect misunderstandings or miscalculations, highlighting the importance of precise arithmetic in scientific estimations.

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2/5 multiplied by ¾ divided by 8/5 equals 316.

6^2 - 3^2 simplifies to 27.

2^3 * 27^(1/3) * 1^3 equals 24.

Extracting the expression results in 5^4 * 2^13.

Approximately 2.5 X 10^5 amoebas are present in a sample that weighs 1 gram.

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