Question

Peter has 50 more baseball cards than his younger brother. After giving his brother 10 cards Peter now has 3 times as many cards as his brother. How many cards does Peter have now

Answer 1

Peter has 45 cards now

SOLUTION

Problem Statement

We are told that Peter has 50 more baseball cards than his younger brother and we are required to find the number of cards Peter has when he gives 10 cards to his younger brother and now has 3 times as many cards as his brother because of the decrease in his number of cards.

Method

- We shall begin by assuming the number of cards Peter has is P and the number of cards that his younger brother has is Y.

- We are told that Peter has 50 more cards than his brother. This means that if we subtract the number of cards Peter's brother has from the number of cards Peter has, we will find that Peter has 50 more cards. Expressed Mathematically, we have:

`P-Y=50\text{ (Equation 1)}`

- We are also told that Peter gives his brother 10 cards and the result of this is that Peter now has 3 times as many cards as his brother. We can break this statement down into two to make sense of it mathematically. This is done below:

`\begin{gathered} \text{Peter gives his brother 10 cards:} \\ \text{Number of cards with Peter after giving 10 cards out= }P-10 \\ \text{ Number of cards with Peter's younger brother after receiving 10 cards = }Y+10 \\ \\ \text{Peter now has 3 times as many cards as his brother:} \\ \text{After Peter has given the cards out, we have that:} \\ \text{ Number of cards with Peter }=\text{ Number of cards with Peter's younger brother }*3 \\ P-10=(Y+10)*3 \\ \text{Simplifying the expression, we have:} \\ P-10=3Y+30 \\ \text{ Subtract 3Y from both sides and add 10 to both sides, we have:} \\ P-10+10-3Y=3Y-3Y+30+10 \\ \\ \therefore P-3Y=40\text{ (Equation 2)} \end{gathered}`

Now that we have represented the statements in the question as a pair of equations, we can solve the equations simultaneously.

This will enable us to find the number of cards Peter had initially, which will, in turn, help us find the number of cards he has now.

Implementation

`\begin{gathered} P-Y=50\text{ (Equation 1)} \\ P-3Y=40\text{ (Equation 2)} \\ \\ \text{Subtract Equation 2 from Equation 1, we have:} \\ P-Y-(P-3Y)=50-40_{} \\ P-Y-P+3Y=10 \\ P-P+3Y-Y=10 \\ 2Y=10 \\ \text{Divide both sides by 2} \\ (2Y)/(2)=(10)/(2) \\ \\ \therefore Y=5 \\ \text{Thus, Peter's younger brother had 5 cards initially.} \\ \\ \text{Substitute the value of Y into Equation 1 to find the value of P} \\ P-Y=50 \\ Y=5, \\ P-5=50 \\ \text{Add 5 to both sides} \\ P=50+5 \\ \therefore P=55 \\ \text{Thus, Peter had 55 cards initially.} \end{gathered}`

From the above calculation, we can see that Peter had 55 cards initially. But we are asked to find the number of cards Peter has after giving out 10 cards to his brother. This means that we decrease Peter's 55 cards by 10 cards. That is:

`\begin{gathered} \text{ Number of cards with Peter now is:} \\ (55-10)\text{cards}=45\text{cards} \end{gathered}`

Thus, Peter has 45 cards now

Final Answer

Peter has 45 cards now