Solving Dina's Coin Problem A Step-by-Step Math Guide

by Scholario Team 54 views

Hey guys! Today, we're diving into a super fun math problem that involves coins, money, and a little bit of detective work. It's like a mini-mystery, and we're the detectives! Our main goal is to break down the problem step-by-step, so you can see how easy it can be to solve these types of questions. We'll be using a mix of simple math and a bit of logical thinking. So, grab your thinking caps, and let's get started!

Unpacking the Coin Conundrum: Dina's Dilemma

So, what's the big puzzle we're tackling today? Here it is: Dina has a total of 20 coins. These coins are a mix of Rp500 coins and Rp1,000 coins. Now, here's the juicy part: the total value of all these coins is Rp15,000. Our mission, should we choose to accept it (and we totally do!), is to figure out exactly how many of each type of coin Dina has. Is it more Rp500 coins, or are there more of the shiny Rp1,000 ones? That's what we need to find out!

Why This Coin Problem Rocks

Before we jump into solving, let's quickly chat about why this kind of problem is actually pretty awesome. First off, it's super practical. We deal with money every day, right? Whether it's buying snacks, saving up for something cool, or even just understanding how much change we should get back, money math is a real-world skill. This problem helps us think about value, quantity, and how different amounts can add up.

Secondly, it's a fantastic brain workout. It's not just about memorizing formulas; it's about using logic and reasoning. We need to figure out how the number of coins and their values connect. It's like flexing our mental muscles!

Our Detective Toolkit: What We'll Use to Solve

Okay, so we know what the problem is, and we know why it's cool. Now, what tools are we going to use to crack this case? We'll be using a few key math concepts, but don't worry, it's nothing too scary. We're mainly talking about:

  • Variables: These are like placeholders for the unknown stuff. We'll use letters (like x and y) to represent the number of Rp500 and Rp1,000 coins.
  • Equations: These are like math sentences. They show how different things are related. We'll build equations to show the relationship between the number of coins and their total value.
  • System of Equations: Because we have two unknowns (the number of each type of coin), we'll actually need two equations. This is called a system of equations, and it's like having two clues that work together.

Think of it like this: we have a bunch of puzzle pieces (the numbers and information in the problem), and our toolkit helps us put them together to see the whole picture.

Cracking the Code: Setting Up the Equations

Alright, detectives, it's time to roll up our sleeves and get those equations going! Remember, equations are just math sentences that help us describe the relationships in the problem. We've got two main clues to work with: the total number of coins and the total value of the money.

Equation #1: The Coin Count

Let's start with the easier one: the total number of coins. We know Dina has 20 coins in total. Some are Rp500, and some are Rp1,000. So, we can say:

  • Let 'x' be the number of Rp500 coins.
  • Let 'y' be the number of Rp1,000 coins.

Now, we can write our first equation: x + y = 20

This equation simply says that the number of Rp500 coins (x) plus the number of Rp1,000 coins (y) equals the total number of coins, which is 20. Easy peasy, right?

Equation #2: The Money Matters

Okay, now for the slightly trickier one: the total value of the money. We know Dina has Rp15,000 in total. To figure out how to write this as an equation, we need to think about the value of each type of coin.

  • Each Rp500 coin is worth Rp500, so 'x' number of Rp500 coins is worth 500x.
  • Each Rp1,000 coin is worth Rp1,000, so 'y' number of Rp1,000 coins is worth 1000y.

Now, we can write our second equation: 500x + 1000y = 15000

This equation says that the total value of the Rp500 coins (500x) plus the total value of the Rp1,000 coins (1000y) equals the total value of all the coins, which is Rp15,000.

Our System is Go!

Boom! We've got our two equations. This is what we call a system of equations:

  1. x + y = 20
  2. 500x + 1000y = 15000

Now, we have a system of equations that captures all the important information from the problem. Solving this system will give us the values of 'x' and 'y', which will tell us how many of each type of coin Dina has. The next step is to actually solve these equations. Exciting, right?

Unlocking the Solution: Methods to the Madness

Alright, team, we've got our system of equations locked and loaded. Now comes the fun part: actually solving them! There are a couple of cool ways we can tackle this, and I'll walk you through the most popular ones. Think of these methods as different tools in our detective toolkit – each one can help us crack the case, but they work in slightly different ways.

Method 1: Substitution – The Name-Swapping Trick

The substitution method is like a clever name-swapping trick. The basic idea is to solve one equation for one variable (say, get 'x' all by itself), and then substitute that expression into the other equation. This lets us get rid of one variable and have an equation with just one unknown, which is much easier to solve.

Let's see how it works with our Dina's coin problem:

  1. Pick an equation and solve for one variable: Let's take our first equation, x + y = 20, and solve for x. To do this, we simply subtract y from both sides: x = 20 - y. Now we know that x is the same as 20 - y.
  2. Substitute: Now, we'll take this expression for x (20 - y) and substitute it into our second equation (500x + 1000y = 15000). This means we replace the 'x' in the second equation with '(20 - y)'. Our equation now looks like this: 500(20 - y) + 1000y = 15000
  3. Solve the new equation: Now we have an equation with just 'y', which we can solve. First, we distribute the 500: 10000 - 500y + 1000y = 15000. Then, we combine like terms: 10000 + 500y = 15000. Next, subtract 10000 from both sides: 500y = 5000. Finally, divide both sides by 500: y = 10. Yay! We've found that y (the number of Rp1,000 coins) is 10.
  4. Back-substitute: We're not done yet! We know y = 10, but we still need to find x. This is where the