Jonathan collects postcards and stamps. The number of postcards in his collection is 12 more than [tex]$\frac{3}{4}$[/tex] the number of stamps. He has 39 postcards in all. If Jonathan has [tex]$x$[/tex] stamps, which equation represents this situation and how many stamps does he have?
A. The equation is [tex]$\frac{3}{4} x + 12 = 39$[/tex], and the number of stamps is 36.
B. The equation is [tex]$\frac{3}{4} x + x = 39$[/tex], and the number of stamps is 24.
C. The equation is [tex]$x + \frac{3}{4} + x = 39$[/tex], and the number of stamps is 18.
D. The equation is [tex]$\frac{3}{4} x = 39 + x$[/tex], and the number of stamps is 156.
Let's carefully analyze the information in the problem to determine the correct equation and the number of stamps Jonathan has.
Step 1: Understand the relationship between postcards and stamps.
Jonathan collects both postcards and stamps. According to the problem, the number of postcards he has is 12 more than [tex]\(\frac{3}{4}\)[/tex] the number of stamps.
Let [tex]\(x\)[/tex] represent the number of stamps Jonathan has.
The number of postcards he has can be expressed as: [tex]\[\frac{3}{4}x + 12\][/tex]
Step 2: Set up the equation based on the given total number of postcards.
We are told that Jonathan has 39 postcards in all. This can be represented by the equation: [tex]\[\frac{3}{4}x + 12 = 39\][/tex]
Step 3: Solve the equation for [tex]\(x\)[/tex].
To find the value of [tex]\(x\)[/tex], we need to isolate [tex]\(x\)[/tex] on one side of the equation.
1. Start with the equation: [tex]\[\frac{3}{4}x + 12 = 39\][/tex]
2. Subtract 12 from both sides to isolate the term with [tex]\(x\)[/tex]: [tex]\[\frac{3}{4}x = 39 - 12\][/tex] [tex]\[\frac{3}{4}x = 27\][/tex]
3. To solve for [tex]\(x\)[/tex], multiply both sides by the reciprocal of [tex]\(\frac{3}{4}\)[/tex], which is [tex]\(\frac{4}{3}\)[/tex]: [tex]\[x = 27 \times \frac{4}{3}\][/tex] [tex]\[x = 27 \times \frac{4}{3} = 36\][/tex]
So, Jonathan has 36 stamps.
Step 4: Check the options for the correct answer.
We see that option A corresponds to the equation [tex]\(\frac{3}{4}x + 12 = 39\)[/tex] and states that the number of stamps is 36, which matches our solution.
Thus, the correct answer is:
A. The equation is [tex]\(\frac{3}{4} x + 12 = 39\)[/tex], and the number of stamps is 36.
