Question
Which description is paired with its correct expression?
four less than the quotient of a number cubed and seven, increased by three; 4 minus StartFraction n cubed Over 7 EndFraction + 3
five times the difference of a number squared and six; 5 (6 minus n squared)
nine more than the quotient of six and a number cubed, decreased by four; 9 + StartFraction n cubed Over 6 EndFraction minus 4
twice the difference of nine and a number squared; 2 (9 minus n squared)
four less than the quotient of a number cubed and seven, increased by three; 4 minus StartFraction n cubed Over 7 EndFraction + 3
five times the difference of a number squared and six; 5 (6 minus n squared)
nine more than the quotient of six and a number cubed, decreased by four; 9 + StartFraction n cubed Over 6 EndFraction minus 4
twice the difference of nine and a number squared; 2 (9 minus n squared)
Answers
GPT-4o mini
Let's analyze each description and check if they are paired with their corresponding expression correctly:
1. **Description:** "four less than the quotient of a number cubed and seven, increased by three"
**Expression:** \(4 - \frac{n^3}{7} + 3\)
**Correct Interpretation:** The "quotient of a number cubed and seven" is \(\frac{n^3}{7}\), and "four less than that" is \(\frac{n^3}{7} - 4\), then "increased by three" would be \(\frac{n^3}{7} - 4 + 3\), which simplifies to \(\frac{n^3}{7} - 1\). So this description is **not correct** with the provided expression as it does not match the intended interpretation.
2. **Description:** "five times the difference of a number squared and six"
**Expression:** \(5(6 - n^2)\)
**Correct Interpretation:** "the difference of a number squared and six" should be \(n^2 - 6\), and then five times that would be \(5(n^2 - 6)\), which is \(5n^2 - 30\). Therefore, this description is **not correct** with the provided expression.
3. **Description:** "nine more than the quotient of six and a number cubed, decreased by four"
**Expression:** \(9 + \frac{n^3}{6} - 4\)
**Correct Interpretation:** The "quotient of six and a number cubed" is \(\frac{6}{n^3}\), so "nine more than that" would be \(9 + \frac{6}{n^3}\) and then "decreased by four" keeps that structure intact, giving us \(9 + \frac{6}{n^3} - 4\), which simplifies to \(5 + \frac{6}{n^3}\). Since the expression given in the pair is not the same as interpreted, this description is **not correct** either.
4. **Description:** "twice the difference of nine and a number squared"
**Expression:** \(2(9 - n^2)\)
**Correct Interpretation:** The phrase "the difference of nine and a number squared" means \(9 - n^2\), and if we multiply that by two, we get \(2(9 - n^2)\), which is correctly matched to the expression. Therefore, this description is **correct**.
The correct pairing is:
**Twice the difference of nine and a number squared; \(2(9 - n^2)\)**.
1. **Description:** "four less than the quotient of a number cubed and seven, increased by three"
**Expression:** \(4 - \frac{n^3}{7} + 3\)
**Correct Interpretation:** The "quotient of a number cubed and seven" is \(\frac{n^3}{7}\), and "four less than that" is \(\frac{n^3}{7} - 4\), then "increased by three" would be \(\frac{n^3}{7} - 4 + 3\), which simplifies to \(\frac{n^3}{7} - 1\). So this description is **not correct** with the provided expression as it does not match the intended interpretation.
2. **Description:** "five times the difference of a number squared and six"
**Expression:** \(5(6 - n^2)\)
**Correct Interpretation:** "the difference of a number squared and six" should be \(n^2 - 6\), and then five times that would be \(5(n^2 - 6)\), which is \(5n^2 - 30\). Therefore, this description is **not correct** with the provided expression.
3. **Description:** "nine more than the quotient of six and a number cubed, decreased by four"
**Expression:** \(9 + \frac{n^3}{6} - 4\)
**Correct Interpretation:** The "quotient of six and a number cubed" is \(\frac{6}{n^3}\), so "nine more than that" would be \(9 + \frac{6}{n^3}\) and then "decreased by four" keeps that structure intact, giving us \(9 + \frac{6}{n^3} - 4\), which simplifies to \(5 + \frac{6}{n^3}\). Since the expression given in the pair is not the same as interpreted, this description is **not correct** either.
4. **Description:** "twice the difference of nine and a number squared"
**Expression:** \(2(9 - n^2)\)
**Correct Interpretation:** The phrase "the difference of nine and a number squared" means \(9 - n^2\), and if we multiply that by two, we get \(2(9 - n^2)\), which is correctly matched to the expression. Therefore, this description is **correct**.
The correct pairing is:
**Twice the difference of nine and a number squared; \(2(9 - n^2)\)**.