Deducing the Value of ( frac{abc}{def} ) from Given Ratios

In this article, we will explore the process of solving a mathematical puzzle involving ratios. We are given the following relationships between variables:

( frac{a}{b} frac{1}{3} ) ( frac{b}{c} 2 ) ( frac{c}{d} frac{1}{2} ) ( frac{d}{e} 3 ) ( frac{e}{f} frac{1}{4} )

Our goal is to determine the value of ( frac{abc}{def} ). We will solve this step-by-step, expressing each variable in terms of ( a ) and then substituting these expressions into the target equation.

Step 1: Express Each Variable in Terms of ( a )

We start by expressing each variable in terms of ( a ) using the given ratios:

From ( frac{a}{b} frac{1}{3} ), we have ( b 3a ). From ( frac{b}{c} 2 ), we can express ( c ) as ( c frac{b}{2} frac{3a}{2} ). From ( frac{c}{d} frac{1}{2} ), we have ( d 2c 2 cdot frac{3a}{2} 3a ). From ( frac{d}{e} 3 ), we can express ( e ) as ( e frac{d}{3} frac{3a}{3} a ). From ( frac{e}{f} frac{1}{4} ), we have ( f 4e 4a ).

Step 2: Calculate ( frac{abc}{def} )

Now we can substitute these expressions into ( frac{abc}{def} ):

[ abc a cdot b cdot c a cdot 3a cdot frac{3a}{2} frac{9a^3}{2} ]

[ def d cdot e cdot f 3a cdot a cdot 4a 12a^3 ]

Substituting these values into ( frac{abc}{def} ) gives:

[ frac{abc}{def} frac{frac{9a^3}{2}}{12a^3} frac{9}{2 cdot 12} frac{9}{24} frac{3}{8} ]

Thus, the value of ( frac{abc}{def} ) is ( frac{3}{8} ).

Alternative Method: Using Simple Fraction Operations

We can also solve this problem using the relationships directly to simplify the expression:

From the given ratios, we can write:

[ frac{a}{d} frac{a}{b} cdot frac{b}{c} cdot frac{c}{d} frac{1}{3} cdot 2 cdot frac{1}{2} frac{1}{3} ]

[ frac{b}{e} frac{b}{c} cdot frac{c}{d} cdot frac{d}{e} 2 cdot frac{1}{2} cdot 3 3 ]

[ frac{c}{f} frac{c}{d} cdot frac{d}{e} cdot frac{e}{f} frac{1}{2} cdot 3 cdot frac{1}{4} frac{3}{8} ]

Therefore, substituting these simplifications into the expression ( frac{abc}{def} ) gives:

[ frac{abc}{def} frac{1}{3} cdot 3 cdot frac{3}{8} frac{3}{8} ]

This confirms the earlier result, showing the value of ( frac{abc}{def} ) is ( frac{3}{8} ).

Conclusion

The value of ( frac{abc}{def} ) is ( frac{3}{8} ), as determined through both detailed substitutions and simplified fraction operations. Understanding and manipulating ratios in this way can help solve a variety of mathematical problems efficiently.