Sam had 2 boxes of cookies.Box A contained 55 more cookies than Box B at first.Sam transferred 310 cookies from Box A to Box B.In the end,the ratio of the number of cookies in Box B to the number of bookies in Box A was 6:1.
a)How many fewer cookies were there in Box A than in Box B in the end?
b)How many cookies were there in the two cookies altogether?
Let's use algebra to solve this problem.
1. Let's start by setting up some variables:
- Let B = the number of cookies in Box B at first
- Then A = B + 55 (since Box A had 55 more cookies than Box B at first)
2. After Sam transfers 310 cookies from Box A to Box B:
- Box B now has B + 310 cookies
- Box A now has A - 310 cookies, which is (B + 55) - 310
3. The problem tells us that the ratio of cookies in Box B to Box A is 6:1 in the end. We can set up an equation for this:
- (B + 310) / (B + 55 - 310) = 6/1
4. Now we can solve for B:
- Cross-multiplying gives us (B + 310) = 6(B - 255)
- Expanding the brackets and simplifying gives us 4B = 2190
- So B = 547.5 (we get a decimal here, but remember that we can't have half a cookie, so we'll round up to the nearest whole number)
5. We can use this value of B to find A:
- A = B + 55 = 602.5 (rounding up again)
6. Now we can answer the questions:
a) How many fewer cookies were there in Box A than in Box B in the end?
- Box A had 602.5 cookies and Box B had 547.5 cookies, so there were 55 fewer cookies in Box B than in Box A in the end.
b) How many cookies were there in the two boxes altogether?
- Box A had 602.5 cookies and Box B had 547.5 cookies, so together they had 1150 cookies.
Let's break down the problem step by step:
Step 1: Let's assume the initial number of cookies in Box B as 'x'. Since Box A has 55 more cookies than Box B, the initial number of cookies in Box A would be 'x + 55'.
Step 2: Sam transfers 310 cookies from Box A to Box B. Therefore, the updated number of cookies in Box A is 'x + 55 - 310' = 'x - 255', and the updated number of cookies in Box B is 'x + 310'.
Step 3: The ratio of the number of cookies in Box B to the number of cookies in Box A is given as 6:1. We can write this as (x + 310)/(x - 255) = 6/1.
Step 4: We can cross-multiply to solve for x: 6(x - 255) = (x + 310).
Step 5: Expanding the equation, we have 6x - 1530 = x + 310.
Step 6: Combining like terms, we get 6x - x = 310 + 1530.
Step 7: Simplifying, we have 5x = 1840.
Step 8: Dividing by 5 on both sides, we find x = 368.
Now that we know the value of 'x', we can find the answers to the questions:
a) The number of cookies in Box A in the end is 'x - 255' = 368 - 255 = 113.
The number of cookies in Box B in the end is 'x + 310' = 368 + 310 = 678.
Therefore, the difference in the number of cookies between Box A and Box B is 678 - 113 = 565.
b) The total number of cookies in both boxes is 'x - 255 + x + 310' = 2x + 55 = 2(368) + 55 = 791.