The reciprocal of a number is 1 divided by the number. The sign does not change.

Turn the fraction upside down.

\(1 \div (-\frac{7}{8}) = -\frac{8}{7}\)

Write both fractions with the lowest common denominator of 2 and 8, which is 8. The denominators then match, so the numerators can be added:

\(\frac{ 1 }{ 2 } + \frac{ 3 }{ 8 } = \frac{ 4 }{ 8 } + \frac{ 3 }{ 8 } = \frac{ 7 }{ 8 }\)

The lowest common denominator of 6 and 12 is 12.

\(\frac{ 5 }{ 6 } - \frac{ 1 }{ 12 } = \frac{ 10 }{ 12 } - \frac{ 1 }{ 12 } = \frac{ 9 }{ 12 }\)

9 and 12 both divide by 3:

\(\frac{ 9 }{ 12 } = \frac{3}{4}\)

Adding \(\frac{ 1 }{ 8 }\) and \(\frac{ 3 }{ 4 }\) is not adding 1 to 3 and 8 to 4.

A denominator says how many equal parts one whole is cut into, so it does not change when parts are collected together. Only fractions written over the same denominator can have their numerators added.

The lowest common denominator of 8 and 4 is 8:

\(\frac{ 1 }{ 8 } + \frac{ 3 }{ 4 } = \frac{ 1 }{ 8 } + \frac{ 6 }{ 8 } = \frac{ 7 }{ 8 }\)

Multiply the numerators, then multiply the denominators:

\(\frac{ 9 }{ 10 } \times \frac{ 1 }{ 6 } = \frac{ 9 \times 1 }{ 10 \times 6 } = \frac{ 9 }{ 60 }\)

9 and 60 both divide by 3:

\(\frac{ 9 }{ 60 } = \frac{ 3 }{ 20 }\)

Count the squares in the grid: \(6\) squares, which is \(12\) halves.

The 3 whole shaded squares count as \(3 \times 2 = 6\) halves, and the 2 half shaded squares count as 2 halves.

Shaded: \(6 + 2 = 8\) halves out of \(12\) halves.

\(\frac{ 8 }{ 12 } = \frac{ 2 }{ 3 }\)

Add the whole numbers first: \(3 + 4 = 7\)

The lowest common denominator of 4 and 2 is 4:

\(\frac{ 3 }{ 4 } + \frac{ 1 }{ 2 } = \frac{ 3 }{ 4 } + \frac{ 2 }{ 4 } = \frac{ 5 }{ 4 }\)

That is more than one whole, so exchange: \(\frac{ 5 }{ 4 } = 1\frac{ 1 }{ 4 }\)

\(7 + 1\frac{ 1 }{ 4 } = 8\frac{1}{4}\)

Write each mixed number as an improper fraction:

\(4\frac{1}{6} = \frac{ 25 }{ 6 }\) and \(2\frac{1}{2} = \frac{ 5 }{ 2 }\)

Multiply the numerators and the denominators:

\(\frac{ 25 }{ 6 } \times \frac{ 5 }{ 2 } = \frac{ 125 }{ 12 }\)

Dividing 125 by 12 gives 10 with 5 left over:

\(\frac{ 125 }{ 12 } = 10\frac{5}{12}\)

Dividing by a fraction is the same as multiplying by its reciprocal — the fraction turned upside down:

\(\frac{ 7 }{ 10 } \div \frac{ 7 }{ 8 } = \frac{ 7 }{ 10 } \times \frac{ 8 }{ 7 } = \frac{ 56 }{ 70 }\)

56 and 70 both divide by 14:

\(\frac{ 56 }{ 70 } = \frac{4}{5}\)

Write each mixed number as an improper fraction:

\(3\frac{5}{6} = \frac{ 23 }{ 6 }\) and \(1\frac{2}{3} = \frac{ 5 }{ 3 }\)

Turn the second fraction upside down and multiply:

\(\frac{ 23 }{ 6 } \times \frac{ 3 }{ 5 } = \frac{ 69 }{ 30 }\)

69 and 30 both divide by 3:

\(\frac{ 69 }{ 30 } = \frac{ 23 }{ 10 } = 2\frac{3}{10}\)

Leave the fraction with \(b\) on its own and write the right-hand side over 150:

\(\frac{ 1 }{ b } = \frac{ 17 }{ 30 } - \frac{ a }{ 5 } = \frac{ 85 - 30a }{ 150 }\)

Call the top \(m = 85 - 30a\). Cross-multiplying gives \(b \times m = 150\), so \(b = \frac{ 150 }{ m }\) is a whole number only when \(m\) divides 150.

\(\frac{ a }{ 5 }\) is less than \(\frac{ 17 }{ 30 }\), so \(a\) is a whole number no bigger than 2. Trying each \(a\) from 1 upwards gives \(m\) = 55, 25 in turn.

Only \(m = 25\) divides 150, and it comes from \(a = 2\). So \(a = 2\) and \(b = \frac{ 150 }{ 25 } = 6\).

Check: \(\frac{ 2 }{ 5 } + \frac{ 1 }{ 6 } = \frac{ 12 }{ 30 } + \frac{ 5 }{ 30 } = \frac{ 17 }{ 30 }\).

A line needs a third number that makes the product 1, so that number is the reciprocal of the product of the other two. Start with a blank that shares a line with two numbers already in the grid.

A is in the top row, with \(\frac{1}{5}\) and \(125\). Their product is \(\frac{1}{5} \times 125 = 25\), so A is the reciprocal of \(25\):

\(A = \frac{1}{25}\)

B is in the left column, with \(\frac{1}{5}\) and \(25\). \(\frac{1}{5} \times 25 = 5\), so B is the reciprocal of this:

\(B = \frac{1}{5}\)

C is in the middle row, with \(\frac{1}{5}\) and \(1\). \(\frac{1}{5} \times 1 = \frac{1}{5}\), so C is the reciprocal of this:

\(C = 5\)

Check: the other lines through A, B and C also multiply to 1.