The ratio of quarters to dimes in a coin collection is 5:3. You add the same number of new quarters as dimes to the collection.
a. Is the ratio of quarters to dimes still 5:3?
b. If so, illustrate your answer with an example. If not, show why with a counter-example.
A. No
B. Assume you have 5 quarters and 3 dimes. Add any amount to that to see what happens.
To determine whether the ratio of quarters to dimes in the coin collection remains 5:3 after adding the same number of new quarters as dimes, we can use algebraic reasoning.
Let's assume that the original number of quarters in the collection is represented by 5x, where x is a positive integer representing the ratio multiplier. Similarly, the original number of dimes can be represented by 3x.
a. To find out if the ratio remains 5:3, let's calculate the number of new quarters and dimes added. Since the same number of new quarters and dimes are added, we can represent this number as x. So, the collection now has (5x + x) quarters and (3x + x) dimes.
Simplifying these expressions, we get:
Quarters: 6x
Dimes: 4x
Now, comparing the new ratio of quarters to dimes, we have 6x:4x. Dividing both sides by 2 to simplify further, we get 3x:2x. This ratio is not the same as the original ratio of 5:3.
b. To illustrate this, let's assume the original collection had 10 quarters (5x) and 6 dimes (3x). If we add 10 new quarters and 6 new dimes, the expanded collection would have 20 quarters (10 original + 10 new) and 12 dimes (6 original + 6 new).
The new ratio of quarters to dimes is 20:12, which simplifies to 5:3. This example shows that the ratio of quarters to dimes can remain the same after adding the same number of new quarters as dimes.
In conclusion, if the original ratio of quarters to dimes is 5:3, the ratio can remain unchanged if the same number of new quarters and dimes are added.