Chan Owen
10/07/2023 · Primary School
If \( \frac{a}{b}=\frac{1}{3}, \frac{b}{c}=2, \frac{c}{d}=\frac{1}{2}, \frac{d}{e}=3 \neq \frac{e}{f}=\frac{1}{4} \) then what is the value of \( \frac{a b c}{\text { def }} \) ?
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Step-by-step Solution
To find the value of \( \frac{a b c}{\text{def}} \), we will first express \( a, b, c, d, e, \) and \( f \) in terms of a single variable.
Starting with the given ratios:
1. From \( \frac{a}{b} = \frac{1}{3} \), we can express \( a \) in terms of \( b \):
\[
a = \frac{1}{3}b
\]
2. From \( \frac{b}{c} = 2 \), we can express \( b \) in terms of \( c \):
\[
b = 2c
\]
3. From \( \frac{c}{d} = \frac{1}{2} \), we can express \( c \) in terms of \( d \):
\[
c = \frac{1}{2}d
\]
4. From \( \frac{d}{e} = 3 \), we can express \( d \) in terms of \( e \):
\[
d = 3e
\]
5. From \( \frac{e}{f} = \frac{1}{4} \), we can express \( e \) in terms of \( f \):
\[
e = \frac{1}{4}f
\]
Now, we will substitute back to express all variables in terms of \( f \).
Starting from \( e \):
\[
e = \frac{1}{4}f
\]
Substituting \( e \) into the equation for \( d \):
\[
d = 3e = 3 \left(\frac{1}{4}f\right) = \frac{3}{4}f
\]
Substituting \( d \) into the equation for \( c \):
\[
c = \frac{1}{2}d = \frac{1}{2} \left(\frac{3}{4}f\right) = \frac{3}{8}f
\]
Substituting \( c \) into the equation for \( b \):
\[
b = 2c = 2 \left(\frac{3}{8}f\right) = \frac{3}{4}f
\]
Substituting \( b \) into the equation for \( a \):
\[
a = \frac{1}{3}b = \frac{1}{3} \left(\frac{3}{4}f\right) = \frac{1}{4}f
\]
Now we have all variables expressed in terms of \( f \):
- \( a = \frac{1}{4}f \)
- \( b = \frac{3}{4}f \)
- \( c = \frac{3}{8}f \)
- \( d = \frac{3}{4}f \)
- \( e = \frac{1}{4}f \)
Next, we calculate \( \frac{abc}{def} \):
\[
abc = \left(\frac{1}{4}f\right) \left(\frac{3}{4}f\right) \left(\frac{3}{8}f\right) = \frac{1 \cdot 3 \cdot 3}{4 \cdot 4 \cdot 8} f^3 = \frac{9}{128} f^3
\]
\[
def = \left(\frac{3}{4}f\right) \left(\frac{1}{4}f\right) \left(f\right) = \frac{3 \cdot 1 \cdot 1}{4 \cdot 4 \cdot 1} f^3 = \frac{3}{16} f^3
\]
Now we can find \( \frac{abc}{def} \):
\[
\frac{abc}{def} = \frac{\frac{9}{128} f^3}{\frac{3}{16} f^3} = \frac{9}{128} \cdot \frac{16}{3} = \frac{9 \cdot 16}{128 \cdot 3} = \frac{144}{384} = \frac{3}{8}
\]
Thus, the value of \( \frac{abc}{def} \) is:
\[
\boxed{\frac{3}{8}}
\]
Quick Answer
The value of \( \frac{abc}{def} \) is \( \frac{3}{8} \).
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