Question
A certain recipe for rice and beans calls for 22 cups of beans and 77 cups of rice. Write a part-to-part ratio of beans to rice. Then write a part-to-whole ratio of beans to the total cups of rice and beans. Finally, write a fraction in simplest terms to express the proportion of beans to the whole.
Answers
GPT-4o mini
To find the various ratios and fractions for the given recipe:
1. **Part-to-part ratio of beans to rice:**
The part-to-part ratio compares the quantities of beans to rice directly. We have 22 cups of beans and 77 cups of rice. Thus, the part-to-part ratio is:
\[
\text{Beans : Rice} = 22 : 77
\]
This ratio can be simplified. Both numbers can be divided by 11:
\[
22 \div 11 = 2, \quad 77 \div 11 = 7 \quad \Rightarrow \quad 2 : 7
\]
So, the simplified part-to-part ratio is **2 : 7**.
2. **Part-to-whole ratio of beans to the total cups of rice and beans:**
To find the total amount of rice and beans, we add the two quantities together:
\[
\text{Total} = 22 \text{ (beans)} + 77 \text{ (rice)} = 99 \text{ cups}
\]
The part-to-whole ratio of beans to the total is:
\[
\text{Beans : Total} = 22 : 99
\]
This ratio can be simplified by dividing both terms by 11:
\[
22 \div 11 = 2, \quad 99 \div 11 = 9 \quad \Rightarrow \quad 2 : 9
\]
Therefore, the part-to-whole ratio of beans to the total is **2 : 9**.
3. **Fraction in simplest terms to express the proportion of beans to the whole:**
The fraction representing the proportion of beans to the total can be written as:
\[
\text{Proportion} = \frac{\text{Beans}}{\text{Total}} = \frac{22}{99}
\]
This fraction can also be simplified by dividing both the numerator and denominator by 11:
\[
\frac{22 \div 11}{99 \div 11} = \frac{2}{9}
\]
Thus, the proportion of beans to the whole in simplest terms is **\(\frac{2}{9}\)**.
In summary:
- Part-to-part ratio of beans to rice: **2 : 7**
- Part-to-whole ratio of beans to total: **2 : 9**
- Fraction representing the proportion of beans to the whole: **\(\frac{2}{9}\)**.
1. **Part-to-part ratio of beans to rice:**
The part-to-part ratio compares the quantities of beans to rice directly. We have 22 cups of beans and 77 cups of rice. Thus, the part-to-part ratio is:
\[
\text{Beans : Rice} = 22 : 77
\]
This ratio can be simplified. Both numbers can be divided by 11:
\[
22 \div 11 = 2, \quad 77 \div 11 = 7 \quad \Rightarrow \quad 2 : 7
\]
So, the simplified part-to-part ratio is **2 : 7**.
2. **Part-to-whole ratio of beans to the total cups of rice and beans:**
To find the total amount of rice and beans, we add the two quantities together:
\[
\text{Total} = 22 \text{ (beans)} + 77 \text{ (rice)} = 99 \text{ cups}
\]
The part-to-whole ratio of beans to the total is:
\[
\text{Beans : Total} = 22 : 99
\]
This ratio can be simplified by dividing both terms by 11:
\[
22 \div 11 = 2, \quad 99 \div 11 = 9 \quad \Rightarrow \quad 2 : 9
\]
Therefore, the part-to-whole ratio of beans to the total is **2 : 9**.
3. **Fraction in simplest terms to express the proportion of beans to the whole:**
The fraction representing the proportion of beans to the total can be written as:
\[
\text{Proportion} = \frac{\text{Beans}}{\text{Total}} = \frac{22}{99}
\]
This fraction can also be simplified by dividing both the numerator and denominator by 11:
\[
\frac{22 \div 11}{99 \div 11} = \frac{2}{9}
\]
Thus, the proportion of beans to the whole in simplest terms is **\(\frac{2}{9}\)**.
In summary:
- Part-to-part ratio of beans to rice: **2 : 7**
- Part-to-whole ratio of beans to total: **2 : 9**
- Fraction representing the proportion of beans to the whole: **\(\frac{2}{9}\)**.