A phrase is shown.
"three times the quantity five less than $x$, divided by the product of six and $x$"
Which expression is equivalent to this phrase?
A. $\frac{3 x-5}{6 x}$
B. $\frac{3 x-5}{x+6}$
C. $\frac{3(x-5)}{6 x}$
D. $\frac{3(x-5)}{6} \cdot x$
Answer (1)
Translates 'five less than x ' to x − 5 .
Translates 'three times the quantity five less than x ' to 3 ( x − 5 ) .
Translates 'the product of six and x ' to 6 x .
Combines these to form the expression 6 x 3 ( x − 5 ) , which matches option C. The final answer is 6 x 3 ( x − 5 ) .
Explanation
Understanding the Problem We need to translate the given phrase into a mathematical expression and then choose the correct option.
Translating the Phrase The phrase is 'three times the quantity five less than x , divided by the product of six and x '.
'Five less than x ' translates to x − 5 .
'Three times the quantity five less than x ' translates to 3 ( x − 5 ) .
'The product of six and x ' translates to 6 x .
'Divided by' means we form a fraction.
Forming the Expression Therefore, the entire phrase translates to the expression 6 x 3 ( x − 5 ) .
Comparing with Options Now we compare this expression with the given options:
A. 6 x 3 x − 5 B. x + 6 3 x − 5 C. 6 x 3 ( x − 5 ) D. 6 3 ( x − 5 ) ⋅ x
The expression we derived, 6 x 3 ( x − 5 ) , matches option C.
Final Answer The equivalent expression is 6 x 3 ( x − 5 ) .
Examples
Understanding how to translate phrases into mathematical expressions is a fundamental skill in algebra. For example, if you are calculating the cost of items with a discount, you might say 'the price after a 20% discount', which translates to 0.8 × original price . Similarly, if you're determining how much to save each month to reach a goal, you might express it as 'total savings divided by the number of months'. These translations help in setting up equations to solve real-world problems.
