Paola L.
asked • 08/27/19Diagramming Ratios
John and Amy had the same amount of money at first. After John spent $8.75 and Amy spent $29.25, the ratio of John's money to Amy's money was 5:3. How much money did Amy have at first? I'm supposed to solve this problem with a diagram and not equations since I'm on 6th grade
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1 Expert Answer
Let the amount of money that both John & Amy have = x
Then (x - 8.75)/(x-29.25) = 5/3 or 3x - 26.25 = 5x -146.25 or 2x = 120
That means that x = amount both have at the beginning = $60
Check: (60 - 8.75)/(60 - 29.25) = 1.67 and 5/3 = 1.67
Paola L.
Thank you so much. Unfortunately, I can't use equations because I'm in sixth grade and we are only doing pre algebra. I have to diagram the problem but I'm having a really hard time.
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08/27/19
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Sarah J.
I know that you must have turned this in long ago, but hopefully, this response will help other people with the same question understand how to diagram and solve this problem, especially since this is a "Challenge Question" in the exercise, so it isn't really demonstrated in the textbook. REMEMBER TO TAKE TIME TO ASK YOUR TEACHER TO CLARIFY THIS! Asking for help from reliable sources (usually not the internet) is an important part of learning! But, here is an answer, on the internet, that I believe to be reliable. Answering this without the ability to put a diagram in this explanation will be tricky...but, let's use our imaginations and visualization skills and see where we get. We are going to represent John's and Amy's portions of the ratio by 2 rows of equal size boxes, one row which represents John's portion of the ratio, placed above another row which will represent Amy's portion of the ratio, as demonstrated in your textbook as a way to diagram ratios. Let's start with what we do know. John and Amy started with the same amount. That is what makes the rest of this possible. We also know that the ratio of John's to Amy's "after spending amounts" (which I will refer to as their "after amount") is 5:3, so John's "after amount" can be represented by 5 boxes, and Amy's "after amount" can be represented by 3 boxes (as you have seen demonstrated in your textbook). I find this best to do on graph paper, as it helps me keep my boxes exact and helps me visualize the differences here, but as long as you are careful, your diagram should help you visualize this scenario on any sort of paper. Remember that each square in the rows represents 1 unit of the ratio. We know that John spent $8.75 to get to his after amount and Amy spent $29.25 to get to her after amount. Since we know that they started with the same amount of money, we can find what the dollar difference between what Amy's and John's "after amounts" is by finding the difference between how much John spent and how much Amy spent to arrive at their "after amounts" that happen to have a 5:3 ratio. We can find that amount of difference by subtracting the amount John spent from the amount that Amy spent ($29.25-$8.75) to get a difference of $20.50. So, Amy spent the same amount as John ($8.75), plus another $20.50 more than John to get to their after amounts (remember that they started with the same amount). Now we know that the current amount of Amy's "after amount" is $20.50 less than John's, and John's after amount is $20.50 more than Amy's. Look at your diagram. See how many units of difference there are between John's portion of the ratio (5 units) and Amy's portion of the ratio (3 units)? It should be clear in your diagram that there are 2 units of difference between John's and Amy's after amounts. Go ahead and bracket those 2 units of the difference shown on John's amount (or the empty boxes in Amy's row, if you tracked the difference that way), and label it with the amount of difference that we discovered: $20.50. If 2 units of the ratio equal $20.50, we can see that 1 unit would be half of $20.50. We can divide the $ amount of difference by the number of units that make up that number ($20.50 divided by 2), to see that each unit (represented by each box in our diagram) is $10.25. It will probably help you get through the next steps if you write down what 1 unit equals on your diagram. If Amy's after amount is 3 units of $10.25, those 3 units are worth $30.75 (3x$10.25). If you take her after amount of $30.75, and add it to what we were told she spent, $29.25, then we get $60. The amount that Amy started with is $60. To check our work, we can make sure John's after amount of $51.25 (5x10.25) plus the amount he spent, ($8.95) is the same as Amy's, since they explained that John and Amy started with the same amount of money. Since $51.25 (what we figured John's after amount is according to the ratio units we discovered) + $8.95 is equal to $60, this shows us that our answer is probably right. We can check out work again, another way. To be extra sure, let's take $60 (the amount we have decided they probably started with if our math is correct) and find John's after spending amount ($60-$8.75), which is $51.25, and Amy's after spending amount, ($60-$29.25) of $30.75. (remember that they told us how much they spent to get to the 5:3 ratio). If divide Jonh's amount by 5, the number of units he has in the after amount ratio ($51.25/5) to get $10.25, and double check that by comparing it to Amy's after amount divided by 3, the number of units of the after amount ratio that she has ($30.75/3), to get $10.25. Since they are the same, we can be very confident that we found the correct numbers. Whew! That one is a doozy. Good luck to all others working this out.12/13/20