In A Bag, The Ratio Of Purple Counters To Green Counters Was 3 : 4 3:4 3 : 4 .8 More Purple Counters Were Then Added To The Bag, And The Ratio Of Purple Counters To Green Counters Became 5 : 4 5:4 5 : 4 .How Many Green Counters Are In The Bag?

Mar 10, 2025 by ADMIN 244 views

In a Bag, the Ratio of Purple Counters to Green Counters Was $3:4$

The problem states that the ratio of purple counters to green counters in a bag was $3:4$ . This means that for every $3$ purple counters, there were $4$ green counters. To better understand this ratio, let's assume that the number of purple counters is $3x$ and the number of green counters is $4x$ , where $x$ is a positive integer.

It is given that $0.8$ more purple counters were added to the bag. This means that the new number of purple counters is $3x + 0.8$ . The ratio of purple counters to green counters became $5:4$ , which means that for every $5$ purple counters, there are $4$ green counters.

Let's set up an equation to represent the new ratio. We know that the number of purple counters is $3x + 0.8$ and the number of green counters is still $4x$ . The ratio of purple counters to green counters is $5:4$ , so we can set up the following equation:

\frac{3x + 0.8}{4x} = \frac{5}{4}

To solve the equation, we can start by cross-multiplying:

4(3x + 0.8) = 5(4x)

Expanding the equation, we get:

12x + 3.2 = 20x

Subtracting $12x$ from both sides, we get:

3.2 = 8x

Dividing both sides by $8$ , we get:

x = 0.4

Now that we have found the value of $x$ , we can find the number of green counters. We know that the number of green counters is $4x$ , so we can substitute $x = 0.4$ into the equation:

\text{Number of green counters} = 4x = 4(0.4) = 1.6

However, the number of green counters must be a whole number, as we cannot have a fraction of a counter. This means that our initial assumption that the number of purple counters is $3x$ and the number of green counters is $4x$ must be incorrect.

Let's revisit the initial ratio of purple counters to green counters, which was $3:4$ . This means that for every $3$ purple counters, there were $4$ green counters. To find the number of green counters, we can set up an equation using the ratio:

\frac{\text{Number of purple counters}}{\text{Number of green counters}} = \frac{3}{4}

Let's assume that the number of purple counters is $3x$ and the number of green counters is $4x$ . We know that $0.8$ more purple counters were added to the bag, so the new number of purple counters is $3x + 0.8$ . The ratio of purple counters to green counters became $5:4$ , so we can set up the following equation:

\frac{3x + 0.8}{4x} = \frac{5}{4}

To solve the equation, we can start by cross-multiplying:

4(3x + 0.8) = 5(4x)

Expanding the equation, we get:

12x + 3.2 = 20x

Subtracting $12x$ from both sides, we get:

3.2 = 8x

Dividing both sides by $8$ , we get:

x = 0.4

However, this value of $x$ does not give us a whole number of green counters. Let's try a different approach.

Let's use the ratio of purple counters to green counters to find the number of green counters. We know that the ratio is $3:4$ , so for every $3$ purple counters, there are $4$ green counters. Let's assume that the number of purple counters is $3x$ and the number of green counters is $4x$ .

We know that $0.8$ more purple counters were added to the bag, so the new number of purple counters is $3x + 0.8$ . The ratio of purple counters to green counters became $5:4$ , so we can set up the following equation:

\frac{3x + 0.8}{4x} = \frac{5}{4}