24. If [tex]a: b=3: 5[/tex] and [tex]a: c=5: 7[/tex], what is the value of [tex]b: c[/tex]?
a) 20
b) 24
c) [tex]21: 25[/tex]
a) [tex]3: 7[/tex]
b) [tex]5: 7[/tex]
What is its diameter?
Answer (1)
Express the given ratios as fractions: b a = 5 3 and c a = 7 5 .
Solve for a in both equations: a = 5 3 b and a = 7 5 c .
Equate the two expressions for a : 5 3 b = 7 5 c .
Solve for the ratio c b : c b = 21 25 . The correct answer is 25 : 21 .
Explanation
Understanding the Problem We are given the ratios a : b = 3 : 5 and a : c = 5 : 7 . Our goal is to find the ratio b : c .
Expressing Ratios as Fractions First, let's express the given ratios as fractions: b a = 5 3 and c a = 7 5
Solving for a Now, we can solve for a in both equations: a = 5 3 b and a = 7 5 c
Equating the Expressions Since both expressions are equal to a , we can set them equal to each other: 5 3 b = 7 5 c
Solving for b/c Now, we solve for the ratio c b :
c b = 7 5 ⋅ 3 5 = 21 25
Finding the Answer Therefore, the ratio b : c is 25 : 21 . However, we need to check if this ratio is among the given options. The options are: a) 20 b) 24 c) 21 : 25 d) 3 : 7 e) 5 : 7 Since 25 : 21 is not directly among the options, let's examine the options more closely. Option c) is 21 : 25 , which is the inverse of our calculated ratio. However, we are looking for b : c , not c : b . Since none of the options match 25 : 21 , there might be an error in the problem statement or the provided options. However, if we assume that the question meant to ask for c : b , then the answer would be 21 : 25 . But since we are looking for b : c , the correct answer should be 25 : 21 , which is not among the options. There seems to be a typo in the options. The correct ratio should be 25 : 21 .
Conclusion Since none of the provided options exactly match the correct ratio b : c = 25 : 21 , and the closest option is 21 : 25 , which represents c : b , we must conclude that there is an error in the options. The correct answer is b : c = 25 : 21 .
Examples
Ratios are used in everyday life, such as when cooking. For example, if a recipe calls for a ratio of flour to sugar of 3:5, and you want to use 25 grams of sugar, you can use ratios to determine that you need 15 grams of flour. Ratios are also used in scaling recipes up or down, maintaining the correct proportions of ingredients to ensure the dish tastes as expected. Understanding ratios helps in many practical situations, from cooking to mixing chemicals in a lab.
