Let's carefully analyze the provided descriptions to determine which equation correctly represents the given statement:
Written Description:
"Eight times the difference between 23.7 and a number is equal to 76.4."
This sentence can be broken down as follows:
1. Identify the difference: "the difference between 23.7 and a number" can be represented mathematically by [tex]\( 23.7 - m \)[/tex], where [tex]\( m \)[/tex] is the number.
2. Eight times the difference: To express "eight times the difference," we multiply the expression by 8. Therefore, [tex]\( 8 \times (23.7 - m) \)[/tex].
3. Is equal to 76.4: We set the equation from step 2 equal to 76.4 resulting in [tex]\( 8 \times (23.7 - m) = 76.4 \)[/tex].
Given these analyses, let's compare the step-by-step interpretations with the provided equations:
1. Equation: [tex]\( 8(m - 23.7) = 76.4 \)[/tex]
- This represents "eight times the difference between a number and 23.7 is equal to 76.4" which is not exactly stated in the given description.
2. Equation: [tex]\( 8m - 23.7 = 76.4 \)[/tex]
- This translates as "eight times a number minus 23.7 is equal to 76.4," which differs from the given description.
3. Equation: [tex]\( 8(23.7 - m) = 76.4 \)[/tex]
- This expression matches our step-by-step breakdown perfectly: "eight times the difference between 23.7 and a number is equal to 76.4."
4. Equation: [tex]\( 8 \times (23.7) - m = 76.4 \)[/tex]
- This represents "eight times 23.7 minus a number is equal to 76.4," which also does not match the description.
Based on the given written description and the step-by-step analysis:
The correct equation that represents the description "Eight times the difference between 23.7 and a number is equal to 76.4" is:
[tex]\[ 8(23.7 - m) = 76.4 \][/tex]
Therefore, the equation that represents the description is the third one:
[tex]\[ 8(23.7 - m) = 76.4 \][/tex]
