Problem Solving with Linear Equations... If she earns a total of $74, how many lemonades does she sell?

Question

Theresa sells lemonade for 1 USD, juice for 1.50 USD,and coffee for 2 USD. The number of coffees she sells is twice the number of lemonades she sells, and 4 more than the number of juices she sells. If she earns a total of 74 USD, how many lemonades does she sell?

I know the answer is 10 lemonades but how could that answer be true? I have tried multiple ways to form an equation but I cannot seem to find the correct one. I don't know if I should make an equation. Ex, the coffee=2x. Juice= 1.50+(x+4). Lemonade=x. If someone just could translate the word problems into an equation, that would be more than just enough of a help!

From the help I received (The other comments solved for Lemonade and for Juice), so I tried to solve for coffee and then substitute the result for coffee in the first equation. Equation 1: 1L+1.5J+2C=74. And we know that C= 2L so it would mean L= C/2. And we also know C=J+4 so it would mean J= C-4. We substitute C/2 for lemonade and C-4 for Juice. And we get: (C/2)+2C+1.5(C-4)=74. So we get: (2C2)+C/2 + 1.5(C-4)=74 into 4C+C/2 + 1.5(C-4)=74. We simplify (5C) numerator and (2) denominator by 2 and get 2.5C+1.5C-6=74 and collect like terms. 4C-6=74 (we add 6 to both sides.) 4c=80 (Divide both sides by 4.) C=20. So finally we can (20/2) + 2(20) + 1.5 (20-4)=74. Lemonade= 10. Coffee= 20. Juice= 16

Answer 1

The most challenging part of word problems is the translation into equations. That is the part you need to learn. This takes time and practice, and reading and re-reading word problems many times. You have to learn what phrases mean as equations.

Phrases like "thing1 is twice as big as thing2" mean that if thing2 is times by 2 we would get thing1, we would write $$T1=2\cdot T2$$ Here's another "there are 4 more apples than pears", as an equation $$A=4+P$$

After reading the entire word problem, I see that Theresa is trying to sell three things and has made some money. I see she is trying to sell lemonade, juice, and coffee. We see she made \$74. Does it make sense this translates to an equation? $$1\cdot L+1.5\cdot J+2\cdot C = 74$$

Now the more tricky parts, "coffees is twice the number of lemonades", do you see this as $$C = 2\cdot L$$ Now we translate "coffees is 4 more than the number of juices". Do you see the $$C = 4+J$$

Sometimes it helps to rephrase things in your own words. "I have 4 more apples than pears", "so if I add 4 to the number of pears I have, I have the amount off apples I have", $A=4+P$. This takes practice. Notice, I don't attempt to translate everything at once, I think about each phrase one at a time.

This is now the easy part, since we have translated all the words into equations, I no longer have to think about the words. I can simply focus on how to do algebra. Can you solve it from here amd find the number of lemonades that were sold? $$1\cdot L+1.5\cdot J+2\cdot C = 74$$ $$C = 2\cdot L$$ $$C = 4+J$$

Answer 2

Let $c$ be the number of coffee she sells; $j$ be the number of juices she sells; and $l$ be the number of lemonade she sells. From this problem we can see three equations based on one statement concerning the relationships between them:

1. The number of coffees she sells is twice the number of lemonades she sells, and 4 more than the number of juices she sells. $$c=2l,\;\;\;\;c=j+4.$$ Since the total is 74, we can write the following: $l+1.5j+2c = 74.$ We can then substitute for $c$, since we know the number of cups of coffee in relation to the number of cups juice she sells. Therefore, we have the following: $l+1.5j+2(j+4) = 74.$
   The next step is to distribute, $l+1.5j+2j+8=74.$ Now we have two variables that is unknown to us. Let's read the statement again and see what we can find:

The number of coffees she sells is twice the number of lemonades she sells, and 4 more than the number of juices she sells.

This means that $2l$ is the same as $j+4$. But we do not want $2l$, we want to find $l$. Again, since $j+4$ is the same thing as $2l$, we can divide $j+4$ by $2$. Once we do that, we can solve the equation. $\frac{j+4}{2}+1.5j+2j+8=74.$ Now we can multiply everything by 2 since that is the common denominator. $j+4+3j+4j+16=148.$ Then, combine like terms. $8j+20=148,\;\;$ $8j+20-20=148-20,\;\;$ $8j=128,\;\;$ $\frac{8j}{8}=\frac{128}{8},\;\;$ $j=16.$ Remember that $j$ is the juice variable. Once we solved for j, we can find $l$. $l+1.5j+2j+8=74.$ This time, we will substitute 16 for $j$ and solve for $l$. $l+1.5(16)+2(16)+8=74,\;\;$ $l+24+32+8=74,\;\;$ $l+64=74,\;\;$ $l+64-64=74-64,\;\;$ $l=10.$ Therefore, she sold 10 cups of lemonade.

Answer 3

Let $l$ be the number of lemonades which Theresa sells; let $j$ be the number of juices she sells; let $c$ be the number of coffees she sells. We were told that each lemonade sells for $\$1$, that each juice sells for $\$1.50$, and that each coffee sells for $\$2$. We were also told that her total sales were $\$74$. Hence, $$l + 1.5j + 2c = 74 \tag{1}$$ In addition, we were told that the number of coffees Theresa sold is twice the number of lemonades that she sold, so $$c = 2l$$ and that the number of coffees she solds was four more than the number of juices she sold, so $$c = j + 4$$ Hence, $$j = c - 4 = 2l - 4$$ If we substitute $2l$ for $c$ and $2l - 4$ for $j$ in equation $1$, we obtain $$l + 1.5(2l - 4) + 2(2l) = 74 \tag{2}$$ Solving equation $2$ for $l$ will give you the number of lemonades she sells.

You can check your work in the following way. After you determine $l$, use the equations $c = 2l$ and $j = 2l - 4$ to find, respectively, the number of coffees and juices that Theresa sold. Substitute your values for $l$, $j$, and $c$ into the expression $l + 1.5j + 2c$. If you solved the problem correctly, you should find that $l + 1.5j + 2c = 74$.