Calculate Shake Prices: Algebra Word Problem
Hey guys, ever found yourself staring at a menu, wondering how the prices of those delicious shakes are calculated? Sometimes, it's not as straightforward as you'd think, especially when different sizes are involved. Today, we're diving deep into a classic word problem that will test your algebra skills and maybe even help you budget your next milkshake run. We'll be tackling a scenario involving small, medium, and large shakes, and using the power of mathematics to uncover their individual prices. So, grab your notebooks, sharpen those pencils, and let's break down this problem piece by piece.
Setting Up the Equations: Translating Words into Math
Alright, let's get down to business. The first step in solving any good word problem is to translate the given information into mathematical equations. Think of it like learning a new language – you've got to understand the symbols and how they connect. We're given three key pieces of information, and each one can be turned into an equation. Let's assign variables to the unknown prices: let 's' represent the price of a small shake, 'm' represent the price of a medium shake, and 'l' represent the price of a large shake. This is crucial because once we have these variables, our problem becomes much more manageable. We're essentially creating a system of equations that we can then solve.
Our first statement says: "The price for buying 5 small shakes, 4 medium shakes, and two large shakes is $25." Translating this directly, we get our first equation: 5s + 4m + 2l = 25. This equation represents the total cost of a specific combination of shakes. Keep this one handy; it's going to be a cornerstone of our solution. It's super important to get this right, as one small error here can throw off the entire calculation. So, double-check those numbers and make sure they're correctly represented. Remember, 's', 'm', and 'l' are our unknowns, and we're aiming to find their values.
Now, for the second piece of information: "The price for 3 small shakes, 4 medium shakes, and one large shake is $17." Following the same logic, we can write this as our second equation: 3s + 4m + l = 17. Notice how the coefficients (the numbers in front of the variables) have changed, reflecting a different combination of shakes. This second equation gives us another crucial link in the chain of information we need to solve for 's', 'm', and 'l'. It's like having two different clues that, when put together, start to reveal the bigger picture. Pay close attention to how the quantities of each shake size differ from the first scenario. This is where the algebra really starts to shine, showing us how to handle multiple unknowns and constraints simultaneously.
Finally, we have a relationship between small and medium shakes: "The price of four medium shakes is the same as the price of seven small shakes." This gives us our third equation: 4m = 7s. This equation is a bit different because it only involves two variables, 'm' and 's'. This is a common technique in algebra word problems – sometimes you get a direct relationship between two of your unknowns, which can be a huge help in simplifying the system. This equation tells us something very specific about the pricing structure of the shakes, implying a fixed ratio between the cost of medium and small sizes. We'll definitely be using this relationship to our advantage. So, to recap, we have our system of three equations:
- 5s + 4m + 2l = 25
- 3s + 4m + l = 17
- 4m = 7s
Our mission, should we choose to accept it (and we totally should!), is to solve this system and find the unique values for 's', 'm', and 'l' that satisfy all three conditions. It’s a bit like being a detective, gathering clues and piecing them together to solve a mystery. Ready to crack the case?
Solving the System: The Elimination and Substitution Method
Now that we've got our equations set up, it's time to roll up our sleeves and solve them. There are a few ways to tackle a system of equations like this, but the most common and effective methods are elimination and substitution. We're going to use a combination of both to make our lives easier. The goal is to isolate one variable at a time and then use that information to find the others. It's a step-by-step process, and if you follow along carefully, you'll see how it all unfolds.
Let's start by looking at equations (1) and (2). Notice that both equations have a '4m' term. This is a golden opportunity for the elimination method! If we subtract equation (2) from equation (1), the '4m' terms will cancel each other out, leaving us with an equation that only involves 's' and 'l'. Let's do that subtraction:
(5s + 4m + 2l) - (3s + 4m + l) = 25 - 17
Distributing the negative sign, we get:
5s + 4m + 2l - 3s - 4m - l = 8
Combining like terms:
(5s - 3s) + (4m - 4m) + (2l - l) = 8
This simplifies beautifully to:
2s + l = 8. (Let's call this equation 4)
This is fantastic! We've just reduced our problem from three variables to two. Equation (4) gives us a new relationship between 's' and 'l'. Now, we need to bring in our third equation (3): 4m = 7s. This equation relates 'm' and 's'. We can use this to express 'm' in terms of 's': m = (7/4)s. This is the substitution part kicking in. We're substituting the relationship we found into our other equations.
Now, let's look back at equation (2): 3s + 4m + l = 17. We know from equation (3) that 4m = 7s. We can substitute '7s' directly for '4m' in equation (2)! This will eliminate 'm' from that equation.
So, equation (2) becomes:
3s + (7s) + l = 17
Combining the 's' terms:
10s + l = 17. (Let's call this equation 5)
Now we have two simpler equations, (4) and (5), both involving only 's' and 'l':
- 2s + l = 8
- 10s + l = 17
This is where we can use elimination again! If we subtract equation (4) from equation (5), the 'l' terms will cancel out, allowing us to solve for 's'.
(10s + l) - (2s + l) = 17 - 8
Distributing the negative sign:
10s + l - 2s - l = 9
Combining like terms:
(10s - 2s) + (l - l) = 9
This gives us:
8s = 9
And finally, we can solve for 's':
s = 9/8
So, the price of a small shake is $9/8, which is equal to $1.125. It's a bit of an unusual price, but mathematically, this is what we've found! It's always a good idea to keep fractions until the very end to maintain accuracy. Now that we have the value of 's', we can work backwards to find 'l' and then 'm'. This is the beauty of solving systems of equations – each step builds on the last.
Finding the Prices: Uncovering the Shake Costs
We've successfully found the price of a small shake ('s'), and now it's time to uncover the prices of the medium ('m') and large ('l') shakes. We're in the home stretch, guys, and this is where all our hard work pays off. We'll use the simpler equations we derived to back-substitute our value of 's' and solve for the remaining variables. It’s like putting the final pieces into a puzzle – everything starts to make sense.
Let's use equation (4) to find 'l': 2s + l = 8. We know that s = 9/8. Substituting this value into equation (4):
2 * (9/8) + l = 8
This simplifies to:
18/8 + l = 8
Which further simplifies to:
9/4 + l = 8
To solve for 'l', we subtract 9/4 from both sides:
l = 8 - 9/4
To subtract, we need a common denominator. 8 can be written as 32/4:
l = 32/4 - 9/4
l = (32 - 9) / 4
l = 23/4
So, the price of a large shake is $23/4, which is $5.75. Looking good! We've found the price for two out of the three shake sizes. Now, let's move on to finding the price of a medium shake ('m').
We can use equation (3): 4m = 7s. We know that s = 9/8. Substituting this value into equation (3):
4m = 7 * (9/8)
4m = 63/8
To solve for 'm', we divide both sides by 4 (or multiply by 1/4):
m = (63/8) * (1/4)
m = 63/32
m = 63/32
So, the price of a medium shake is $63/32, which is approximately $1.96875. Again, a bit of an unusual price, but these are the exact mathematical results. To make it easier to understand, we can round this to about $1.97 for practical purposes, but for verification, we'll use the exact fraction.
Verification: Checking Our Answers
Now, the most satisfying part of any math problem: verification! It's always a good idea to plug our calculated prices back into the original equations to make sure everything checks out. This ensures we haven't made any silly mistakes along the way. We found:
s = 9/8 (or $1.125) m = 63/32 (or approx. $1.97) l = 23/4 (or $5.75)
Let's test these values in our original equations:
Equation 1: 5s + 4m + 2l = 25
5 * (9/8) + 4 * (63/32) + 2 * (23/4)
= 45/8 + 252/32 + 46/4
To add these, we need a common denominator, which is 32:
= (454)/32 + 252/32 + (468)/32
= 180/32 + 252/32 + 368/32
= (180 + 252 + 368) / 32
= 800 / 32
= 25
Success! Equation 1 holds true.
Equation 2: 3s + 4m + l = 17
3 * (9/8) + 4 * (63/32) + (23/4)
= 27/8 + 252/32 + 23/4
Using a common denominator of 32:
= (274)/32 + 252/32 + (238)/32
= 108/32 + 252/32 + 184/32
= (108 + 252 + 184) / 32
= 544 / 32
= 17
Nailed it! Equation 2 is also correct.
Equation 3: 4m = 7s
4 * (63/32) = 7 * (9/8)
252/32 = 63/8
Simplify the left side by dividing numerator and denominator by 4:
63/8 = 63/8
Perfect! Equation 3 is satisfied.
All three original equations are satisfied with our calculated values. This confirms that our solution is correct. The price of a small shake is $1.125, a medium shake is approximately $1.97, and a large shake is $5.75. While the medium shake price might seem a little odd in a real-world scenario, mathematically, these are the precise values that fit all the conditions given in the problem. It's a great example of how algebra can help us solve complex, real-world (or menu-world!) problems. Keep practicing these types of problems, guys – the more you do, the more confident you'll become with algebra!