Question
find missing ratio
flour 3 30 15 18
sugar 2 20 10 12
cookies 120 240 60
missing
flour and sugar is missing to 240 cookies
cookies is missing to 3 flour and 2 sugar
cookies is missing on 18 flour and 12 cookies
flour 3 30 15 18
sugar 2 20 10 12
cookies 120 240 60
missing
flour and sugar is missing to 240 cookies
cookies is missing to 3 flour and 2 sugar
cookies is missing on 18 flour and 12 cookies
Answers
Answered by
GPT-4o mini
To find the missing ratios, we can set up proportions based on the data provided.
The structure seems to suggest that the relationship follows the recipe or ratio between flour, sugar, and cookies. Let's break down the information you've provided.
We have:
1. **Flour**: 3 (for ratio) leads to 120 cookies, while 2 (for ratio) leads to 60 cookies, and so on.
2. **Sugar**: Likewise, the given values for sugar correspond to the number of cookies.
3. **Cookies**: The total number of cookies produced is based on the amounts of flour and sugar used.
**1. Flour and Sugar to 240 Cookies:**
Using the ratios given:
- For every 3 parts of flour (which leads to 120 cookies), we need 2 parts of sugar.
If we increase the cookies to 240, which is twice the initial amount (120), we need to double the amount of flour and sugar:
- Flour: **3 parts (for 120 cookies)** to **6 parts (for 240 cookies)**
- Sugar: **2 parts (for 120 cookies)** to **4 parts (for 240 cookies)**
So for **240 cookies**, we need:
- **Flour: 6**
- **Sugar: 4**
**2. Cookies Missing to 3 Flour and 2 Sugar:**
Using the known ratios, for the base recipe:
- 3 parts flour and 2 parts sugar gives:
\[
Cookies = K \cdot Flour^{a} \cdot Sugar^{b}
\]
Where \(K\) is a constant for the specific recipe.
From previous proportions, we know:
- 3 flour and 2 sugar makes **120 cookies** (as 3:2 ratio).
**3. Cookies Missing on 18 Flour and 12 Sugar:**
Now let's check what cookies would be produced with:
- **18 parts of flour** and **12 parts of sugar**.
- From the initial ratio of 3 parts flour (120 cookies) to 2 parts sugar, we can derive the cookies.
The ratio of our recipe (3 flour and 2 sugar) can be reduced:
- If 3 flour and 2 sugar yield 120 cookies, then:
\[
\text{Scale factor for flour} = \frac{18}{3} = 6
\]
\[
\text{Scale factor for sugar} = \frac{12}{2} = 6
\]
Thus, both flour and sugar scaled by the same factor lead to cookies scaled in the same proportion:
- 120 cookies × 6 factor = **720 cookies**.
### To summarize:
- **240 cookies require 6 flour and 4 sugar.**
- **3 flour and 2 sugar yield 120 cookies.**
- **18 flour and 12 sugar yield 720 cookies.**
The structure seems to suggest that the relationship follows the recipe or ratio between flour, sugar, and cookies. Let's break down the information you've provided.
We have:
1. **Flour**: 3 (for ratio) leads to 120 cookies, while 2 (for ratio) leads to 60 cookies, and so on.
2. **Sugar**: Likewise, the given values for sugar correspond to the number of cookies.
3. **Cookies**: The total number of cookies produced is based on the amounts of flour and sugar used.
**1. Flour and Sugar to 240 Cookies:**
Using the ratios given:
- For every 3 parts of flour (which leads to 120 cookies), we need 2 parts of sugar.
If we increase the cookies to 240, which is twice the initial amount (120), we need to double the amount of flour and sugar:
- Flour: **3 parts (for 120 cookies)** to **6 parts (for 240 cookies)**
- Sugar: **2 parts (for 120 cookies)** to **4 parts (for 240 cookies)**
So for **240 cookies**, we need:
- **Flour: 6**
- **Sugar: 4**
**2. Cookies Missing to 3 Flour and 2 Sugar:**
Using the known ratios, for the base recipe:
- 3 parts flour and 2 parts sugar gives:
\[
Cookies = K \cdot Flour^{a} \cdot Sugar^{b}
\]
Where \(K\) is a constant for the specific recipe.
From previous proportions, we know:
- 3 flour and 2 sugar makes **120 cookies** (as 3:2 ratio).
**3. Cookies Missing on 18 Flour and 12 Sugar:**
Now let's check what cookies would be produced with:
- **18 parts of flour** and **12 parts of sugar**.
- From the initial ratio of 3 parts flour (120 cookies) to 2 parts sugar, we can derive the cookies.
The ratio of our recipe (3 flour and 2 sugar) can be reduced:
- If 3 flour and 2 sugar yield 120 cookies, then:
\[
\text{Scale factor for flour} = \frac{18}{3} = 6
\]
\[
\text{Scale factor for sugar} = \frac{12}{2} = 6
\]
Thus, both flour and sugar scaled by the same factor lead to cookies scaled in the same proportion:
- 120 cookies × 6 factor = **720 cookies**.
### To summarize:
- **240 cookies require 6 flour and 4 sugar.**
- **3 flour and 2 sugar yield 120 cookies.**
- **18 flour and 12 sugar yield 720 cookies.**