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Digital SAT Math
Course: Digital SAT Math > Unit 2
Lesson 6: Systems of linear equations word problems: foundationsSystems of linear equations word problems — Harder example
Watch Sal work through a harder Systems of linear equations word problem.
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Agnes has 23 collectible stones, all of which are labradorite crystals or galena crystals. Labradorite crystals are worth $20 each, while galena crystals are worth $13 each. Agnes earns $439 by selling her entire collection. How many stones of each type did she sell?
I'm still stuck on this one problem....please help!(17 votes)
20 labradorite and 3 galena
because 20*20=400 and 3*13=39 so the total will be 439 which is exactly how much Agnes made(22 votes)
good concept but a little bit confusing(25 votes)
Why are we given harder practice problems then what the example shows? This one looks more simple, but the other practice problems are so complicated sometimes. The SAT is coming soon, I just feel so under prepared..(15 votes)- you can solve it with less time and effort. you can substract 4 children and 1 adult from each choice and see if the remaining children are double the adults since each of the remaining adults brought 2 children. think smarter not harder haha(12 votes)
how can we use this for graphs?(6 votes)- Hi, is this problem you immediately knew that there will be only 1 adult with two kids so you did the math of 4+2x2=8.
I thought, however, that there will be two adults (parents) as my fist logical thing to think.
how could one understand that there is only one adult for each 2 kids?
thanks!(4 votes)- Don't overlook "the remaining adults brought 2 children EACH." So, it's children per adult regardless; singles or couples.(5 votes)
- I just solved this question by constructing the following equations and got the same answer
4 + 2(a-1)= c
2c + 4a= 60(5 votes) 
Here is my way of solving it:
We know that they charge $2 for each child and $4 for each adult and the total ticket sales from the children and adults was $60.
2c + 4a = 60
We know that there is an adult that brought 4 children and the remaining adults brought 2 children each.
c = 4 + 2 (a-1)
Let's substitute that into the first equation.
=> 2c + 4a = 60
=> 2 (4 + 2[a-1]) + 4a = 60
=> 2 (4 + 2a - 2) + 4a = 60
=> 2 (2 + 2a) + 4a = 60
=> 4 + 4a + 4a = 60
=> 4 + 8a = 60 Subtract 4 from both side of the equation to isolate the term with the variable
=> 8a = 56 Divide both side of the equation by 8
=> a = 7
Now substitute that information to find out the number of tickets for children.
2c + 4a = 60
2c + 4(7) = 60
2c + 28 = 60 Subtract 28 from both side of the equation
2c = 32 Divide both side of the equation by 2
c = 16(5 votes)- Hello,
Why is 4 + 4 x 2 is 12? If 4 plus 4 = 8, 8 times 2 is 16?(2 votes)- Remember the order of operations. We have to multiply before adding.
4 + 4 x 2
4 + 8
12(4 votes)
- Agnes has 23 collectible stones, all of which are labradorite crystals or galena crystals. Labradorite crystals are worth $20 each, while galena crystals are worth $13 each. Agnes earns $439 by selling her entire collection. How many stones of each type did she sell?
im stuck on this .(1 vote)
This type of problem is pretty common on the SAT, and test to see if you can set up and solve a system of equations. What you want to do is group the information you have into two groups, from which you can create two equations. In this problem, you are given the total number of stones, the price per stone of each stone, and the total price of the collection. Based on the last sentence, you know that your variables should be the number of stones. Everything that talks about the prices seems related. If L is the number of Labradorite crystals and G is the number of galena crystals, you can represent the prices as this:
20L + 13G = 439
This is because multiplying the unit price per crystal by the amount of crystal will give you the money you would get for all the crystals of that type. Add this to the amount of money you would get for the other crystal type and you have the total.
We can set up another equation because we know how many crystals in total Agnes has, which is just x + y = 23.
Now you have the two equations set up, and it's time to solve them. This is a pretty necessary skill for the SAT, so if you don't know it already, I'd recommend you check out some videos on how the process works (https://www.khanacademy.org/math/algebra-basics/alg-basics-systems-of-equations). But, as always on SAT math, setting up this problem and figuring out how to do it is the actual hard part, and the actual doing it is more automatic.
You should end up getting 20 Labradorite crystals and 3 Galena crystals.(5 votes)
Video transcript
- [Instructor] Tickets for a
play were $2 for each child and $4 for each adult. At one showing of the play,
one adult brought four children and the remaining adults
brought two children each. The total ticket sales from the
children and adults was $60. How many children and
adults attended the play? Alright, this is an interesting one. Okay, so let's just think about how much we've spent at the play and we know it has to add up to $60. Let's think about it in
terms of the children and the adults and their admissions. So you have this one adult right over here that brought four children. So how much is that adult,
how much is this family? Let's just assume it's a family. How much are they going to spend? Well, that one adult is going to spend $4 for their own ticket and
then four children at $2 each. So plus four children times $2 per child, this is going to be $8
for the children's tickets plus $4 on theirs. They're going to spend $12. So that adult is going to spend $12. And then there's some
remaining number of adults that brought two children. So let's just say r is I could say the remaining number of adults or the number of adults with two children. Adults with two children, that's r. So each of these adults with two children, how much are they going to spend? Well, they're each going to
spend $4 on their own ticket for the adult and then they're
gonna have two children at $2 each, so they're gonna spend $4 on the children's tickets. So they're gonna spend $8 in total. So each of these adults with two children is gonna spend $8 at the
play and there's r of them. So they're going to spend
$8 for each of these adults with two children and there are r of them. So this is the total
amount of ticket sales from the adults with two children. And we add that to the ticket sales from this one adult with the four children and they're gonna have to add up to $60. So this is gonna have to add up to $60. Let's see, we can subtract
12 from both sides and so on the left, we'll be left with 8r is equal to 60 minus 12 is 48. Divide both sides by eight
and you get r is equal to six. So we wanna be very careful, you might say, okay there was six adults these are just the
adults with two children. There's six adults with two children, but there's another adult. There's another adult who
brought four children. So there's a total of seven adults, seven adults total. Pardon my handwriting. Seven adults total. So we could just look at these choices, only one of these choices
have seven adults. And we could verify that this would also amount to 16 children because this person up here, in magenta, they would bring four children, so you would have four children plus six adults brought two children, so six adults bringing two children each, that would amount to 12 children. And that indeed, that
indeed does add up to be 16. And if you're under time pressure, you can see there is only one
choice that has seven adults, so you could just pick that one.