4) Jack is buying gift bags for his daughter’s 5th birthday party. His budget is between $20 and $30. He has a choice of $2, $3 and $4 bags. If he buys $4 bags he will have $3 remaining, if he buys $3 bags he will have $2 remaining, and if he buys $2 bags he will have $1 remaining. What is
his budget?
I Think if the budget is between 20 and 30 we have to see where he is saving most .That is where the option is to buy $4 bag and still $3 remaining so the budget is $23.
To solve this problem, let's use algebra to find out Jack's budget.
Let's suppose Jack buys x $4 bags, y $3 bags, and z $2 bags.
According to the information given in the question, we can write the following equations:
1) If Jack buys $4 bags, he will have $3 remaining. This can be expressed as 4x + 3 = B (where B is his budget).
2) If Jack buys $3 bags, he will have $2 remaining. This can be expressed as 3y + 2 = B.
3) If Jack buys $2 bags, he will have $1 remaining. This can be expressed as 2z + 1 = B.
From the first equation, we can rearrange it as 4x = B - 3.
From the second equation, we can rearrange it as 3y = B - 2.
From the third equation, we can rearrange it as 2z = B - 1.
Now, let's find the values of x, y, and z that satisfy all the equations simultaneously.
To do this, we can subtract the constant terms on both sides of each equation, which will give us:
4x - B = -3,
3y - B = -2,
2z - B = -1.
Now, let's sum up these equations:
(4x - B) + (3y - B) + (2z - B) = -3 - 2 - 1,
4x + 3y + 2z - 3B = -6.
We can simplify this equation further:
4x + 3y + 2z = 3B - 6.
Since Jack's budget lies between $20 and $30, we know that 20 ≤ B ≤ 30. So we can substitute B with a value between 20 and 30 in the equation above to find the values of x, y, and z.
For example, let's substitute B = 20:
4x + 3y + 2z = 3(20) - 6,
4x + 3y + 2z = 54.
Now, we need to find integer values of x, y, and z that satisfy this equation. We can use trial and error or other algebraic methods to find these values.
Once we have found values for x, y, and z, we can calculate Jack's budget by substituting them into one of the original equations. For example, if we find x = 3, y = 2, and z = 4, we can use the equation 4x + 3y + 2z = B to find:
4(3) + 3(2) + 2(4) = B,
12 + 6 + 8 = B,
26 = B.
Therefore, Jack's budget would be $26.