If [tex]$a: b=3: 5$[/tex] and [tex]$a: c=5: 7$[/tex], what is the value of [tex]$b: c$[/tex]?
A. [tex]$3: 7$[/tex]
B. [tex]$5: 7$[/tex]
C. [tex]$21: 25$[/tex]
D. [tex]$25: 21$[/tex]
Answer (1)
Express the 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 , so b : c = 25 : 21 .
Explanation
Understanding the Problem We are given two ratios: a : b = 3 : 5 and a : c = 5 : 7 . We need to find the ratio b : c .
Expressing Ratios as Fractions We can express the given ratios as fractions: b a = 5 3 and c a = 7 5 .
Solving for a Now, we solve for a in both equations. From the first equation, we have a = 5 3 b . From the second equation, we have 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 by multiplying both sides by 3 5 and dividing by c : c b = 7 5 ⋅ 3 5 = 21 25
Finding the Ratio b:c Therefore, the ratio b : c is 25 : 21 .
Examples
Understanding ratios is crucial in everyday life. For instance, when baking, you might need to adjust ingredient quantities based on a recipe's ratios. If a recipe calls for a flour-to-sugar ratio of 3:5 and you want to use a different amount of flour, you can use ratios to calculate the correct amount of sugar needed to maintain the recipe's balance. Similarly, in construction, ratios are used to determine the correct proportions of materials like cement, sand, and gravel in concrete mixes, ensuring the structure's strength and durability. Ratios help us scale quantities proportionally, whether in cooking, construction, or any field requiring precise adjustments.
