Solving Ratios and Proportions: A Comprehensive Guide
Mathematics often presents us with complex problems that require the thorough understanding and application of basic concepts. One such concept is the ratio and proportion, which is fundamental in various fields, from finance to engineering. This article will delve into solving a specific problem that involves multiple ratios: If A:B2:3, B:C4:5, and C:D6:7, we aim to find the values of A, B, C, and D.
Understanding Ratios and Proportions
In mathematics, a ratio is a comparison of two quantities by division. Proportions, on the other hand, are statements that two ratios are equal. The given problem requires us to understand how to manipulate these ratios to find a common base, which can then be used to determine the values of A, B, C, and D.
Combining Ratios
To solve the problem, we first need to examine the given ratios:
A:B 2:3B:C 4:5C:D 6:7Each ratio represents a relationship between two variables. Our goal is to find a single ratio, A:B:C:D, by making the middle terms (B and C) consistent in all ratios.
Step 1: Aligning B in Both Ratios
To align the B terms, we multiply the first and second ratios by appropriate numbers to make the B terms in both ratios equal. The first ratio is 2:3, and the second ratio is 4:5.
A:B 2:3 can be written as 2*4 : 3*4 8:12B:C 4:5 remains as it is.Thus, the updated ratios now are A:B:C 8:12:15.
Step 2: Aligning C in the Combined Ratio and D
Next, we want to make the C terms consistent in all the ratios. The combined ratio A:B:C 8:12:15, and we need to align this with the last given ratio C:D 6:7. We can achieve this by multiplying the combined ratio by 2 to get A:B:C 16:24:30.
The last given ratio C:D 6:7 can be written as 15*2 : 35*2 30:35. This ensures that the ratios are now consistent.
So, the final combined ratio is A:B:C:D 16:24:30:35.
Direct Calculation Using Common Denominators
Alternatively, we can use the least common multiple (LCM) to find a direct solution:
For the first ratio A:B 2:3, the LCM of 2 and 3 is 6, so A:B 4:6 (2*2:3*2).For the second ratio B:C 4:5, the LCM of 4 and 5 is 20, so B:C 16:20 (4*4:5*4).For the third ratio C:D 6:7, the LCM of 6 and 7 is 42, so C:D 30:35 (6*5:7*5).Thus, A:B:C:D 4:6:10:14, which simplifies to 2:3:5:7.5. Further multiplication by 2 gives A:B:C:D 16:24:30:35.
Conclusion
The values of A, B, C, and D based on the given ratios are A 16, B 24, C 30, and D 35. This detailed approach ensures that we accurately solve the problem by ensuring consistency in the terms of the ratios.