helen and ivan have the same total number of coins. helen has a number of fifty-cent coinsand 64 twenty-cent coins. the total mass of her coins is 1.134 kg. ivan has a number of fifty-cent coins and 104 twenty-cent coins.

a) who has more money in coins?
how much more?

b) Each fifty-cent coin is 2.7g heavier than each twenty-cent coin. what is the total mass of ivan's coins in kg?


give me the answer with proper working method

Let's solve the problem step by step:

### Given Information:
- Helen and Ivan have the same total number of coins.
- Helen has \( x \) fifty-cent coins and 64 twenty-cent coins.
- Ivan has \( y \) fifty-cent coins and 104 twenty-cent coins.
- The total mass of Helen's coins is 1.134 kg.
- Each fifty-cent coin is 2.7g heavier than each twenty-cent coin.

We are tasked with answering two questions:
1. Who has more money in coins? By how much?
2. What is the total mass of Ivan's coins?

### Step 1: Express Helen's Total Mass in Terms of the Coins

Let the mass of a twenty-cent coin be \( m \) grams.

Thus, the mass of a fifty-cent coin is \( m + 2.7 \) grams.

The total mass of Helen's coins is the mass of \( x \) fifty-cent coins and 64 twenty-cent coins. So, her total mass is:
\[
x(m + 2.7) + 64m = 1134 \text{ grams} = 1.134 \text{ kg}.
\]
This simplifies to:
\[
xm + 2.7x + 64m = 1134.
\]
This equation expresses the relationship between the number of coins and their masses.

### Step 2: Determine Ivan's Total Number of Coins

Helen and Ivan have the same total number of coins. Therefore, the number of coins for both is:
\[
x + 64 = y + 104.
\]
Simplifying this:
\[
x - y = 40.
\]
So, Helen has 40 more fifty-cent coins than Ivan.

### Step 3: Solve for \( x \) (Number of Fifty-cent Coins Helen Has)

Substitute \( y = x - 40 \) into the equation for total mass.

To determine the total mass of Ivan's coins and compare their money amounts, we can

helen and ivan have the same total number of coins. helen has a number of fifty-cent coins and 64 twenty-cent coins. the total mass of her coins is 1.134 kg. ivan has a number of fifty-cent coins and 104 twenty-cent coins. a) who has more money in coins? how much more? b) Each fifty-cent coin is 2.7g heavier than each twenty-cent coin. what is the total mass of ivan's coins in kg?

Explore the coin distribution problem involving Helen and Ivan. Solve for who has more money and calculate the mass of Ivan's coins using algebra and weight distribution formulas. Suitable for students in Grades 9-12.

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