If \(\frac{{\rm{a}}}{{\rm{b}}} = \frac{1}{3},\frac{{\rm{b}}}{{\rm{c}}} = 2,\frac{{\rm{c}}}{{\rm{d}}} = \frac{1}{2},\frac{{\rm{d}}}{{\rm{e}}} = 3\) and \(\frac{{\rm{e}}}{{\rm{f}}} = \frac{1}{4}\) , then what is the value of \(\frac{{{\rm{abc}}}}{{{\rm{def}}}}?\)
Calculating Ratio Expressions from Given Ratios
The problem asks us to find the value of the expression \(\frac{{\rm{abc}}}}{{{\rm{def}}}}\) given several individual ratios between consecutive variables: \(\frac{{\rm{a}}}{{\rm{b}}} = \frac{1}{3}\), \(\frac{{\rm{b}}}{{\rm{c}}} = 2\), \(\frac{{\rm{c}}}{{\rm{d}}} = \frac{1}{2}\), \(\frac{{\rm{d}}}{{\rm{e}}} = 3\), and \(\frac{{\rm{e}}}{{\rm{f}}} = \frac{1}{4}\).
Understanding the Expression \(\frac{{{\rm{abc}}}}{{{\rm{def}}}}\)
The expression \(\frac{{{\rm{abc}}}}{{{\rm{def}}}}\) is a fraction with a product of three variables in the numerator and a product of three variables in the denominator. We can rewrite this expression as a product of individual ratios:
\[ \frac{{\rm{abc}}}}{{{\rm{def}}}} = \frac{{\rm{a}}}{{\rm{d}}} \times \frac{{\rm{b}}}{{\rm{e}}} \times \frac{{\rm{c}}}{{\rm{f}}} \]
Our strategy will be to calculate the values of the individual ratios \(\frac{{\rm{a}}}{{\rm{d}}}\), \(\frac{{\rm{b}}}{{\rm{e}}}\), and \(\frac{{\rm{c}}}{{\rm{f}}}}\) using the given ratios, and then multiply these values together.
Step-by-Step Ratio Calculation
Calculating \(\frac{{\rm{a}}}{{\rm{d}}}\)
We can find the ratio \(\frac{{\rm{a}}}{{\rm{d}}}\) by multiplying the consecutive ratios that link 'a' to 'd': \(\frac{{\rm{a}}}{{\rm{b}}}\), \(\frac{{\rm{b}}}{{\rm{c}}}\), and \(\frac{{\rm{c}}}{{\rm{d}}}}\).
\[ \frac{{\rm{a}}}{{\rm{d}}} = \frac{{\rm{a}}}{{\rm{b}}} \times \frac{{\rm{b}}}{{\rm{c}}} \times \frac{{\rm{c}}}{{\rm{d}}} \]
Substitute the given values:
\[ \frac{{\rm{a}}}{{\rm{d}}} = \left(\frac{1}{3}\right) \times (2) \times \left(\frac{1}{2}\right) \]
\[ \frac{{\rm{a}}}{{\rm{d}}} = \frac{1}{3} \times \frac{2}{1} \times \frac{1}{2} = \frac{1 \times 2 \times 1}{3 \times 1 \times 2} = \frac{2}{6} = \frac{1}{3} \]
So, \(\frac{{\rm{a}}}{{\rm{d}}} = \frac{1}{3}\).
Calculating \(\frac{{\rm{b}}}{{\rm{e}}}\)
We can find the ratio \(\frac{{\rm{b}}}{{\rm{e}}}\) by multiplying the consecutive ratios that link 'b' to 'e': \(\frac{{\rm{b}}}{{\rm{c}}}\), \(\frac{{\rm{c}}}{{\rm{d}}}\), and \(\frac{{\rm{d}}}{{\rm{e}}}}\).
\[ \frac{{\rm{b}}}{{\rm{e}}} = \frac{{\rm{b}}}{{\rm{c}}} \times \frac{{\rm{c}}}{{\rm{d}}} \times \frac{{\rm{d}}}{{\rm{e}}} \]
Substitute the given values:
\[ \frac{{\rm{b}}}{{\rm{e}}} = (2) \times \left(\frac{1}{2}\right) \times (3) \]
\[ \frac{{\rm{b}}}{{\rm{e}}} = \frac{2}{1} \times \frac{1}{2} \times \frac{3}{1} = \frac{2 \times 1 \times 3}{1 \times 2 \times 1} = \frac{6}{2} = 3 \]
So, \(\frac{{\rm{b}}}{{\rm{e}}} = 3\).
Calculating \(\frac{{\rm{c}}}{{\rm{f}}}\)
We can find the ratio \(\frac{{\rm{c}}}{{\rm{f}}}\) by multiplying the consecutive ratios that link 'c' to 'f': \(\frac{{\rm{c}}}{{\rm{d}}}\), \(\frac{{\rm{d}}}{{\rm{e}}}\), and \(\frac{{\rm{e}}}{{\rm{f}}}}\).
\[ \frac{{\rm{c}}}{{\rm{f}}} = \frac{{\rm{c}}}{{\rm{d}}} \times \frac{{\rm{d}}}{{\rm{e}}} \times \frac{{\rm{e}}}{{\rm{f}}} \]
Substitute the given values:
\[ \frac{{\rm{c}}}{{\rm{f}}} = \left(\frac{1}{2}\right) \times (3) \times \left(\frac{1}{4}\right) \]
\[ \frac{{\rm{c}}}{{\rm{f}}} = \frac{1}{2} \times \frac{3}{1} \times \frac{1}{4} = \frac{1 \times 3 \times 1}{2 \times 1 \times 4} = \frac{3}{8} \]
So, \(\frac{{\rm{c}}}{{\rm{f}}} = \frac{3}{8}\).
Calculating the Final Expression \(\frac{{{\rm{abc}}}}{{{\rm{def}}}}\)
Now we multiply the values of the ratios we just calculated:
\[ \frac{{\rm{abc}}}}{{{\rm{def}}}} = \frac{{\rm{a}}}{{\rm{d}}} \times \frac{{\rm{b}}}{{\rm{e}}} \times \frac{{\rm{c}}}{{\rm{f}}} \]
Substitute the calculated values:
\[ \frac{{\rm{abc}}}}{{{\rm{def}}}} = \left(\frac{1}{3}\right) \times (3) \times \left(\frac{3}{8}\right) \]
\[ \frac{{\rm{abc}}}}{{{\rm{def}}}} = \frac{1}{3} \times \frac{3}{1} \times \frac{3}{8} = \frac{1 \times 3 \times 3}{3 \times 1 \times 8} = \frac{9}{24} \]
Simplify the fraction \(\frac{9}{24}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 3.
\[ \frac{9 \div 3}{24 \div 3} = \frac{3}{8} \]
So, the value of \(\frac{{\rm{abc}}}}{{{\rm{def}}}}\) is \(\frac{3}{8}\).
| Given Ratio | Value |
|---|---|
| \(\frac{{\rm{a}}}{{\rm{b}}}\) | \(\frac{1}{3}\) |
| \(\frac{{\rm{b}}}{{\rm{c}}}\) | \(2\) |
| \(\frac{{\rm{c}}}{{\rm{d}}}\) | \(\frac{1}{2}\) |
| \(\frac{{\rm{d}}}{{\rm{e}}}\) | \(3\) |
| \(\frac{{\rm{e}}}{{\rm{f}}}\) | \(\frac{1}{4}\) |
| Calculated Ratio | Calculation | Value |
|---|---|---|
| \(\frac{{\rm{a}}}{{\rm{d}}}\) | \(\frac{{\rm{a}}}{{\rm{b}}} \times \frac{{\rm{b}}}{{\rm{c}}} \times \frac{{\rm{c}}}{{\rm{d}}}\) | \(\frac{1}{3} \times 2 \times \frac{1}{2} = \frac{1}{3}\) |
| \(\frac{{\rm{b}}}{{\rm{e}}}\) | \(\frac{{\rm{b}}}{{\rm{c}}} \times \frac{{\rm{c}}}{{\rm{d}}} \times \frac{{\rm{d}}}{{\rm{e}}}\) | \(2 \times \frac{1}{2} \times 3 = 3\) |
| \(\frac{{\rm{c}}}{{\rm{f}}}\) | \(\frac{{\rm{c}}}{{\rm{d}}} \times \frac{{\rm{d}}}{{\rm{e}}} \times \frac{{\rm{e}}}{{\rm{f}}}\) | \(\frac{1}{2} \times 3 \times \frac{1}{4} = \frac{3}{8}\) |
| Final Calculation | Value |
|---|---|
| \(\frac{{\rm{abc}}}}{{{\rm{def}}}} = \frac{{\rm{a}}}{{\rm{d}}} \times \frac{{\rm{b}}}{{\rm{e}}} \times \frac{{\rm{c}}}{{\rm{f}}}\) | \(\frac{1}{3} \times 3 \times \frac{3}{8} = \frac{3}{8}\) |
Revision Table: Key Ratios and Calculations
| Ratio | Value |
|---|---|
| Given Ratios | \(\frac{a}{b}=\frac{1}{3}, \frac{b}{c}=2, \frac{c}{d}=\frac{1}{2}, \frac{d}{e}=3, \frac{e}{f}=\frac{1}{4}\) |
| Intermediate Calculated Ratios | \(\frac{a}{d}=\frac{1}{3}, \frac{b}{e}=3, \frac{c}{f}=\frac{3}{8}\) |
| Final Expression Value | \(\frac{abc}{def} = \frac{a}{d} \times \frac{b}{e} \times \frac{c}{f} = \frac{1}{3} \times 3 \times \frac{3}{8} = \frac{3}{8}\) |
Additional Information on Ratio Operations
When you have a series of ratios linking variables, like \(\frac{a}{b}\) and \(\frac{b}{c}\), you can find the ratio between non-consecutive variables by multiplying the intermediate ratios. For example, \(\frac{a}{c} = \frac{a}{b} \times \frac{b}{c}\). This property is used repeatedly in this problem to find ratios like \(\frac{a}{d}\) or \(\frac{b}{e}\) or \(\frac{c}{f}\).
Also, a complex fraction like \(\frac{abc}{def}\) can often be broken down into simpler fractions or products of fractions involving individual variables or pairs of variables. In this case, rewriting \(\frac{abc}{def}\) as \(\frac{a}{d} \times \frac{b}{e} \times \frac{c}{f}\) (or other combinations like \(\frac{a}{b} \times \frac{b}{c} \times \frac{c}{d} \times \frac{1}{(d/e) \times (e/f)}\) etc.) helps in utilizing the given ratios efficiently. The choice of breakdown \(\frac{a}{d} \times \frac{b}{e} \times \frac{c}{f}\) was convenient because each required ratio could be directly computed by multiplying a set of three consecutive given ratios.
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Match the following.
Column I
Column II
a.
Equivalent fraction of \(\frac{7}{12}\) is
i.
Proper fraction
b.
Equivalent fraction of \(\frac{9}{15}\) is
ii.
Improper fraction
c.
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iii.
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d.
\(\frac{19}{5}\) is
iv.
\(\frac{3}{5}\)
