What is “the quotient of a number and five”, “eight less than twice a number”, and “four times the difference of a number and two”
The Correct Answer and Explanation is:
1. “The quotient of a number and five”
Let the unknown number be represented by the variable x.
The word “quotient” means division.
So, the phrase translates to: x5\frac{x}{5}5x
2. “Eight less than twice a number”
Again, let the number be x.
“Twice a number” means 2 multiplied by the number, or 2x.
“Eight less than” means subtract 8 from the result.
So, the phrase becomes: 2x−82x – 82x−8
3. “Four times the difference of a number and two”
We still use x as the number.
“The difference of a number and two” means subtract 2 from the number: x – 2.
“Four times” this difference means multiply the entire expression by 4.
So, the full expression is: 4(x−2)4(x – 2)4(x−2)
Detailed Explanation
These expressions are examples of how everyday language translates into mathematical symbols. The first phrase, “the quotient of a number and five,” refers to dividing a number by five. In mathematics, division is represented using the division symbol (÷) or a fraction. When we say “the quotient of x and 5,” we write it as x ÷ 5 or more commonly in algebra, as x/5. This is a simple rational expression where the numerator is the unknown and the denominator is 5.
The second phrase, “eight less than twice a number,” is slightly more complex. “Twice a number” means multiplying the unknown by 2, which gives 2x. The phrase “eight less than” tells us to subtract 8 from that product. In mathematics, subtraction is order-sensitive, so 2x – 8 is correct, not 8 – 2x.
The third phrase, “four times the difference of a number and two,” involves both subtraction and multiplication. “The difference of a number and two” becomes x – 2, and “four times” that difference means multiplying the entire difference by 4. In algebra, when multiplying an expression like this, we use parentheses: 4(x – 2). This tells us to first find the difference inside the parentheses and then multiply it by 4.
Understanding how words translate into mathematical expressions helps in solving word problems and writing equations.
