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One morning each member of Angela's family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?

Let $c$ and $m$ denote the total amount of coffee and milk, respectively, in ounces. Then we have the linear system of equations:

$\frac{c}{6} + \frac{m}{4} = 8, \tag1$

$c + m = 8n. \tag2$ Multiplying equation (1) above by $(12)$ gives

$2c + 3m = 96\tag3$ How can I reduce the system to just a one-variable equation solving for $n$? I've tried to manipulate and use substitution in a variety of ways but my experiments don't get me anywhere. I'm just lost/confused at this point.

The problem gave this hint "Let $c$ be the amount of coffee Angela drank, $m$ be the amount of milk she drank, and $n$ be the number of people in the family, Write two equations based on the information in the problem. Slove for $n$ in terms of $c$.

I found letting $c$ be the total amount of coffee and $m$ being the total amount of milk more intuitive. I couldn't figure out the equation for $8n$ otherwise.

The question Find the number of members of a family is similar but the answer presented a formula I haven't studied yet, I'm still on basic algebra.

  • I can not understand the question... Where do we get any information on the other family members' drinking habits? What is preventing us from having a single other family member who drank by themselves all of the milk and coffee that Angela did not? – JMoravitz Apr 13 '21 at 17:39
  • I concur, at the beginning, I felt the problem is incomplete/ ambiguous. That's literally all the information given. It's from "Introduction to Algebra - AOPS" book. –  Apr 13 '21 at 17:43
  • @JMoravitz: it says each one drank a total of 8 ounces – Ross Millikan Apr 13 '21 at 17:43
  • Does it say that all of the liquid is consumed? What is preventing Angela from being the only person in this story and just having not consumed the entirety of the available liquid? Is Angela included in having also necessarily consumed $8$ ounces? – JMoravitz Apr 13 '21 at 17:44
  • I presume "each member of Angela's family drank an 8-ounce mixture of coffee with milk" means they all drink a full 8z mix. –  Apr 13 '21 at 17:46
  • Is it necessary that every family members' drink mixture involves an integer number of ounces of each drink type? It seems to me that this problem statement has many holes in it which require strong assumptions. – JMoravitz Apr 13 '21 at 17:49

1 Answers1

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Your working is correct. Now just solve in terms of $n$ first.

$2c+3m = 96$

$c+m = 8n$

where $n$ is the number of members in Angela's family including her.

Solving both equations, $m = 96 - 16n, c = 24n - 96$

Now which values of $n$ (positive integers) are solutions?

From $m$, you can see $n \lt 6$ and from $c$, $n \gt 4$.

So the only solution is $n = 5$.

Math Lover
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    Ah, I see. I had gotten to $m = 96-16n, c=24n-96$ before but didn't realise that was enough to deduce $n$. I kept subsisting for $m$ or $c$ afterward. Hmm, you really have to pay attention to detail and not just blindly following steps you done before hoping for the best. –  Apr 13 '21 at 18:30