Inches and barleycorns
How many barleycorns are there in one inch?
Age
14 to 16
Challenge level
Problem
Yards, furlongs, inches, cubits, chains and barleycorns are all imperial measures of length.
There are 220 yards in a furlong and 36 inches in a yard. There are 44 cubits in a chain and 54 barleycorns in a cubit. 1 furlong is the same as 10 chains.
How many barleycorns are there in an inch?
This problem is adapted from the World Mathematics Championships
Student Solutions
Answer: 3
Both of the methods shown use diagrams to organise the information given. There are also good methods which do not use diagrams, but involve similar calculations.
Working out how many inches and how many barleycorns there are in a furlong
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$220\times36$ inches = $10\times44\times54$ barleycorns
$1$ inch = $\dfrac{10\times44\times54}{220\times36}=\dfrac{10\times(11\times4)\times(9\times6)}{(11\times2\times10)\times(9\times4)}=3$ barleycorns
Working in inches using a table
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$(54\times44)\div(22\times36) = (6\times9\times4\times11) \div (2\times11\times6\times 6)=3$