# What is the simplest way for a person who has only guineas to pay \[10s.6d.\] to another who has only half-crowns?

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**Hint:**In this question we are given that there is one person who has only guineas and there is another person who has only half-crowns. First person with guineas has to pay \[10s.6d.\] to the second person. To do this first we convert all to the smallest unit penny \[(d)\]. Then we will find the value of guineas such that it can directly be converted to half crowns after subtracting \[10s.6d.\] from half crowns. In this way we can solve this question.

**Formula used:**We have done the conversions between different denominations using the following formulas here,

\[1 \text{guinea} = 252d\]

\[1s = 12d\]

\[1\text{half - crown} = 30d.\]

**Complete step-by-step solution:**

We know that guinea can be converted to penny as,

\[1\text{guinea} = 252d\]

And shilling can be converted as,

\[1s = 12d\]

We are given that we have to give \[10s.6d.\] It can be converted to penny as,

\[

10s.6d. = 10 \times 12 + 6 \\

\Rightarrow 10s.6d. = 126d. \]

We also know the relation between half crown and penny as,

\[1\text{half - crown} = 30d.\]

We know that ,

\[

3\text{guineas} = 3 \times 252d \\

\Rightarrow 3\text{guineas} = 756d \\

\Rightarrow 3\text{guineas} = 630d + 126d \]

Using the relation between half crown and penny, we move ahead as,

\[

\Rightarrow 3\text{guineas}= 21 \times 30d + 126d \\

\Rightarrow 3\text{guineas} = 21\text{half - crowns} + 126d \]

Since, \[126d\] was to be paid already, a person with guineas has to pay \[3\text{guineas}\] and another person has to return \[21\text{half-crowns}\]. This is the simplest way to pay \[10s.6d.\] with only guineas and half-crowns.

**Note:**This is to note that here, we are doing conversions between the British denominations. This is useful when we have to deal in some other denominations and we have currency of completely other denominations. The abbreviations used here are, \[d\] is penny and \[s\] is shilling.

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