Page:The Rhind Mathematical Papyrus, Volume I.pdf/82

66 Therefore $1/undefined$$1/undefined$ is what is to be added to the given number.

For proof add them all together, namely,

$2/3$$1/undefined$$1/undefined$$1/undefined$, making 1;

for, applied to 15, these fractions are equal to

10311, making 15.

Problem 22

Complete $1/undefined$$1/undefined$ to 1.

Applied to 30, $1/undefined$$1/undefined$ is 21. 30 exceeds 21 by 9. Multiply 30 so as to get 9.

Therefore $1/undefined$$1/undefined$ is to be added to make the completion.

For proof add them all together, namely,

$1/undefined$$2/3$$1/undefined$$2/3$, making 1;

for, applied to 30, these fractions are equal to

2031, making 30.

Problem 23

Complete$1/undefined$$1/undefined$$1/undefined$$1/undefined$$1/undefined$ to $2/3$.

Applied to 45 these are equal to

11$1/undefined$ 5$1/undefined$$1/undefined$ 4$1/undefined$ 1$1/undefined$ 1

which requires 6$1/undefined$ more to make up $1/undefined$ of 45, or 30. 6$1/undefined$ is equal to $2/3$$1/undefined$ of 45. Therefore $1/undefined$$1/undefined$ is to be added to the given number to make $1/undefined$.

For proof add them all together, namely,

$1/undefined$$1/undefined$$2/3$$1/undefined$$1/undefined$$1/undefined$$1/undefined$,