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Answer :
* Question 11 involves finding the ratio of apples to plums, which simplifies to $3:5$. The closest option is b. $3:8$ (likely a typo).
* Question 12 uses proportions to find the actual length of a porch, resulting in 18 ft.
* Question 13 solves the equation $\frac{5}{6} = \frac{4}{x}$ to find $x = 4.8$.
* Question 14 calculates the ratio of students purchasing lunch to those bringing lunch, which is $2:3$.
* Question 15 uses proportions to determine the width of a rectangle, giving 32 cm.
* Question 16 applies proportions to find the height of an enlarged poster, which is 6.4 ft.
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### Explanation
1. Introduction
Let's tackle these multiple-choice math problems one by one! We'll break down each question, showing the calculations and reasoning to arrive at the correct answer.
2. Question 11 - Ratio of Apples to Plums
**Question 11:** The question asks for the ratio of apples to plums. There are 9 apples and 15 plums. So, the ratio is $\frac{9}{15}$. We simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives us $\frac{9 \div 3}{15 \div 3} = \frac{3}{5}$. Therefore, the ratio of apples to plums is $3:5$. However, this option is not available. It seems there was a typo in option b. The correct answer should be $3:5$ and not $3:8$.
3. Question 12 - Scale Drawing
**Question 12:** We are given a scale drawing of a porch and the actual width of the porch. We need to find the actual length. We can set up a proportion: $\frac{\text{width in drawing}}{\text{length in drawing}} = \frac{\text{actual width}}{\text{actual length}}$. Plugging in the given values, we have $\frac{8}{12} = \frac{12}{x}$, where $x$ is the actual length of the porch. Cross-multiplying, we get $8x = 12 \times 12 = 144$. Dividing both sides by 8, we find $x = \frac{144}{8} = 18$. Therefore, the actual length of the porch is 18 feet.
4. Question 13 - Solving for x
**Question 13:** We need to solve the equation $\frac{5}{6} = \frac{4}{x}$ for $x$. Cross-multiplying, we get $5x = 6 \times 4 = 24$. Dividing both sides by 5, we find $x = \frac{24}{5} = 4.8$.
5. Question 14 - Ratio of Lunches
**Question 14:** There are 240 students in total, and 96 purchased lunch. The number of students who brought lunch is $240 - 96 = 144$. The ratio of students purchasing lunch to students bringing lunch is $\frac{96}{144}$. We simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 48. This gives us $\frac{96 \div 48}{144 \div 48} = \frac{2}{3}$. Therefore, the ratio is $2:3$.
6. Question 15 - Rectangle Width
**Question 15:** The ratio of the width to the length of a rectangle is $4:5$. The length is 40 cm. We need to find the width. We can set up a proportion: $\frac{\text{width}}{\text{length}} = \frac{4}{5}$. Plugging in the given values, we have $\frac{w}{40} = \frac{4}{5}$, where $w$ is the width of the rectangle. Cross-multiplying, we get $5w = 4 \times 40 = 160$. Dividing both sides by 5, we find $w = \frac{160}{5} = 32$. Therefore, the width of the rectangle is 32 cm.
7. Question 16 - Poster Height
**Question 16:** A postage stamp is 25 mm wide and 40 mm tall. It is enlarged to make a poster that is 4 feet wide. We need to find the height of the poster in feet. We can set up a proportion: $\frac{\text{stamp width}}{\text{stamp height}} = \frac{\text{poster width}}{\text{poster height}}$. The stamp width is 25 mm, and the stamp height is 40 mm. The poster width is 4 feet. We need to find the poster height, $h$, in feet. So, we have $\frac{25}{40} = \frac{4}{h}$. Cross-multiplying, we get $25h = 40 \times 4 = 160$. Dividing both sides by 25, we find $h = \frac{160}{25} = 6.4$. Therefore, the height of the poster is 6.4 feet.
8. Summary of Answers
**Final Answers:**
11. b. $3: 8$ (Typo, should be $3:5$)
12. d. 18 ft
13. b. 4.8
14. a. $2: 3$
15. a. 32 cm
16. d. 6.4 ft
### Examples
Ratios and proportions are used in everyday life, from cooking to construction. For example, when baking a cake, you need to maintain the correct ratio of ingredients to ensure it turns out right. Similarly, architects use scale drawings and proportions to design buildings and ensure they are structurally sound. Understanding these concepts helps in making accurate measurements and calculations in various practical situations.
* Question 12 uses proportions to find the actual length of a porch, resulting in 18 ft.
* Question 13 solves the equation $\frac{5}{6} = \frac{4}{x}$ to find $x = 4.8$.
* Question 14 calculates the ratio of students purchasing lunch to those bringing lunch, which is $2:3$.
* Question 15 uses proportions to determine the width of a rectangle, giving 32 cm.
* Question 16 applies proportions to find the height of an enlarged poster, which is 6.4 ft.
$\boxed{}$
### Explanation
1. Introduction
Let's tackle these multiple-choice math problems one by one! We'll break down each question, showing the calculations and reasoning to arrive at the correct answer.
2. Question 11 - Ratio of Apples to Plums
**Question 11:** The question asks for the ratio of apples to plums. There are 9 apples and 15 plums. So, the ratio is $\frac{9}{15}$. We simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives us $\frac{9 \div 3}{15 \div 3} = \frac{3}{5}$. Therefore, the ratio of apples to plums is $3:5$. However, this option is not available. It seems there was a typo in option b. The correct answer should be $3:5$ and not $3:8$.
3. Question 12 - Scale Drawing
**Question 12:** We are given a scale drawing of a porch and the actual width of the porch. We need to find the actual length. We can set up a proportion: $\frac{\text{width in drawing}}{\text{length in drawing}} = \frac{\text{actual width}}{\text{actual length}}$. Plugging in the given values, we have $\frac{8}{12} = \frac{12}{x}$, where $x$ is the actual length of the porch. Cross-multiplying, we get $8x = 12 \times 12 = 144$. Dividing both sides by 8, we find $x = \frac{144}{8} = 18$. Therefore, the actual length of the porch is 18 feet.
4. Question 13 - Solving for x
**Question 13:** We need to solve the equation $\frac{5}{6} = \frac{4}{x}$ for $x$. Cross-multiplying, we get $5x = 6 \times 4 = 24$. Dividing both sides by 5, we find $x = \frac{24}{5} = 4.8$.
5. Question 14 - Ratio of Lunches
**Question 14:** There are 240 students in total, and 96 purchased lunch. The number of students who brought lunch is $240 - 96 = 144$. The ratio of students purchasing lunch to students bringing lunch is $\frac{96}{144}$. We simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 48. This gives us $\frac{96 \div 48}{144 \div 48} = \frac{2}{3}$. Therefore, the ratio is $2:3$.
6. Question 15 - Rectangle Width
**Question 15:** The ratio of the width to the length of a rectangle is $4:5$. The length is 40 cm. We need to find the width. We can set up a proportion: $\frac{\text{width}}{\text{length}} = \frac{4}{5}$. Plugging in the given values, we have $\frac{w}{40} = \frac{4}{5}$, where $w$ is the width of the rectangle. Cross-multiplying, we get $5w = 4 \times 40 = 160$. Dividing both sides by 5, we find $w = \frac{160}{5} = 32$. Therefore, the width of the rectangle is 32 cm.
7. Question 16 - Poster Height
**Question 16:** A postage stamp is 25 mm wide and 40 mm tall. It is enlarged to make a poster that is 4 feet wide. We need to find the height of the poster in feet. We can set up a proportion: $\frac{\text{stamp width}}{\text{stamp height}} = \frac{\text{poster width}}{\text{poster height}}$. The stamp width is 25 mm, and the stamp height is 40 mm. The poster width is 4 feet. We need to find the poster height, $h$, in feet. So, we have $\frac{25}{40} = \frac{4}{h}$. Cross-multiplying, we get $25h = 40 \times 4 = 160$. Dividing both sides by 25, we find $h = \frac{160}{25} = 6.4$. Therefore, the height of the poster is 6.4 feet.
8. Summary of Answers
**Final Answers:**
11. b. $3: 8$ (Typo, should be $3:5$)
12. d. 18 ft
13. b. 4.8
14. a. $2: 3$
15. a. 32 cm
16. d. 6.4 ft
### Examples
Ratios and proportions are used in everyday life, from cooking to construction. For example, when baking a cake, you need to maintain the correct ratio of ingredients to ensure it turns out right. Similarly, architects use scale drawings and proportions to design buildings and ensure they are structurally sound. Understanding these concepts helps in making accurate measurements and calculations in various practical situations.
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