3. In how many different ways can R20 be obtained from R5; and 20c coins?

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UpStudy AI · Accepted Answer

We want to obtain R20 using only R5 coins and 20¢ coins. To set up the problem, let:

- $$ x $$ be the number of R5 coins.
- $$ y $$ be the number of 20¢ coins.

Since R20 means 20 rupees, and knowing that 1 rupee = 100 cents, we have:
- R5 = $$ 5 \text{ rupees} = 500 \text{ cents} $$.
- 20¢ = $$ 20 \text{ cents} $$.
- R20 = $$ 2000 \text{ cents} $$.

The total value, in cents, can be written as:
$$
500x + 20y = 2000
$$

### Step 1. Simplify the Equation

Divide the entire equation by 10:
$$
50x + 2y = 200
$$

Then divide by 2:
$$
25x + y = 100
$$

### Step 2. Express $$ y $$ in Terms of $$ x $$

Solve for $$ y $$:
$$
y = 100 - 25x
$$

### Step 3. Find the Possible Values for $$ x $$

Since $$ y $$ must be a nonnegative integer:
$$
100 - 25x \geq 0 \quad \Rightarrow \quad 25x \leq 100 \quad \Rightarrow \quad x \leq 4
$$
Also, $$ x $$ must be nonnegative, so $$ x \geq 0 $$.

Thus, $$ x $$ can take the values:
$$
x = 0, 1, 2, 3, 4
$$

### Step 4. Count the Number of Solutions

For each valid value of $$ x $$, there is a corresponding value of $$ y $$:
- If $$ x = 0 $$, then $$ y = 100 - 25(0) = 100 $$.
- If $$ x = 1 $$, then $$ y = 100 - 25(1) = 75 $$.
- If $$ x = 2 $$, then $$ y = 100 - 25(2) = 50 $$.
- If $$ x = 3 $$, then $$ y = 100 - 25(3) = 25 $$.
- If $$ x = 4 $$, then $$ y = 100 - 25(4) = 0 $$.

There are 5 possible pairs $$(x, y)$$.

### Final Answer

There are $$ \boxed{5} $$ different ways to obtain R20 from R5 coins and 20¢ coins.
There are 5 different ways to obtain R20 using R5 coins and 20¢ coins.

3. In how many different ways can R20 be obtained from R5; and 20c coins?

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