Which equation can be used to represent "three minus the difference of a number and one equals one-half of the difference of three times the same number and four"?
$(1-n)-3=\frac{1}{2}(4-3 n)$
$3-(1-n)=\frac{1}{2}(4-3 n)$
$(n-1)-3=\frac{1}{2}(3 n-4)$
$3-(n-1)=\frac{1}{2}(3 n-4)$
Answer (1)
Translate 'three minus the difference of a number and one' to 3 − ( n − 1 ) .
Translate 'one-half of the difference of three times the same number and four' to 2 1 ( 3 n − 4 ) .
Equate the two expressions: 3 − ( n − 1 ) = 2 1 ( 3 n − 4 ) .
The correct equation is 3 − ( n − 1 ) = 2 1 ( 3 n − 4 ) .
Explanation
Problem Analysis We need to translate the given sentence into an algebraic equation. Let's break down the sentence into smaller parts.
Translating the first part The phrase "three minus the difference of a number and one" can be written as 3 − ( n − 1 ) , where n represents the number.
Translating the second part The phrase "one-half of the difference of three times the same number and four" can be written as 2 1 ( 3 n − 4 ) .
Forming the equation The word "equals" means we set the two expressions equal to each other. So, the equation is 3 − ( n − 1 ) = 2 1 ( 3 n − 4 ) .
Finding the correct option Now, let's compare our equation with the given options:
( 1 − n ) − 3 = 2 1 ( 4 − 3 n )
3 − ( 1 − n ) = 2 1 ( 4 − 3 n )
( n − 1 ) − 3 = 2 1 ( 3 n − 4 )
3 − ( n − 1 ) = 2 1 ( 3 n − 4 )
The equation 3 − ( n − 1 ) = 2 1 ( 3 n − 4 ) matches the fourth option.
Final Answer Therefore, the correct equation is 3 − ( n − 1 ) = 2 1 ( 3 n − 4 ) .
Examples
In real life, translating word problems into equations is essential for solving various problems, such as calculating costs, determining distances, or optimizing resources. For example, if you're trying to figure out how much to charge for a product to make a certain profit, you'll need to translate the problem into an equation and solve for the price.
