Verified Answer :
- Define the number of quarters \(Q\) based on the ratio \(A:B\) and dimes \(D\) as \(Q = \frac{D}{A} \times B\).
- Case 3: 15 dimes, 3:2 ratio \(\Rightarrow Q = \frac{15}{3} \times 2 = 10\).
- Case 4: 8 dimes, 1:4 ratio \(\Rightarrow Q = \frac{8}{1} \times 4 = 32\).
- Case 5: 28 dimes, 4:3 ratio \(\Rightarrow Q = \frac{28}{4} \times 3 = 21\).
- Final answers: (3) \(\boxed{10}\), (4) \(\boxed{32}\), (5) \(\boxed{21}\).
Explanation
- Understanding the Ratio and Tape Diagram Approach
We are given the number of dimes and the ratio of dimes to quarters in three separate scenarios. A ratio of \(A:B\) means that for every \(A\) dimes, there are \(B\) quarters. We can represent this with a tape diagram where the ratio represents the number of equal parts. If we have \(D\) dimes, and the ratio of dimes to quarters is \(A:B\), we can determine the value of a single 'part' in our diagram by calculating \(\frac{D}{A}\). Once we have the value of one part, we find the number of quarters by multiplying this value by the number of parts representing quarters, which is \(B\). The formula for the number of quarters \(Q\) is \(Q = \frac{D}{A} \times B\).
- Calculating Quarters for Case 3
In case 3, we have 15 dimes and a ratio of dimes to quarters of \(3:2\).
Applying our formula:
Number of dimes \(D = 15\).
Ratio \(A:B = 3:2\).
Value of one part = \(\frac{15}{3} = 5\).
Number of quarters \(Q = 5 \times 2 = 10\).
So, for 15 dimes with a ratio of 3:2, there are 10 quarters.
- Calculating Quarters for Case 4
In case 4, we have 8 dimes and a ratio of dimes to quarters of \(1:4\).
Applying our formula:
Number of dimes \(D = 8\).
Ratio \(A:B = 1:4\).
Value of one part = \(\frac{8}{1} = 8\).
Number of quarters \(Q = 8 \times 4 = 32\).
So, for 8 dimes with a ratio of 1:4, there are 32 quarters.
- Calculating Quarters for Case 5
In case 5, we have 28 dimes and a ratio of dimes to quarters of \(4:3\).
Applying our formula:
Number of dimes \(D = 28\).
Ratio \(A:B = 4:3\).
Value of one part = \(\frac{28}{4} = 7\).
Number of quarters \(Q = 7 \times 3 = 21\).
So, for 28 dimes with a ratio of 4:3, there are 21 quarters.
- Summary of Results
We have successfully determined the number of quarters for each of the three cases:
- Case 3: 10 quarters
- Case 4: 32 quarters
- Case 5: 21 quarters
Thus, the answers are:
(3) 10, (4) 32, (5) 21.
Examples
Understanding ratios and tape diagrams is essential in everyday life when mixing ingredients or adjusting recipes. For example, if a recipe calls for a ratio of 3 cups of flour to 2 cups of sugar, and you have 15 cups of flour, you can use the same tape diagram method to find out you need 10 cups of sugar to keep the proportions correct. This ensures that your baking or any other proportional task turns out exactly as intended!