(a) Simplify \(\frac{3}{m + 2n} - \frac{2}{m - 3n}\)
(b) A number is made up of two digits. The sum of the digits is 11. If the digits are interchanged, the original number is increased by 9. Find the number.
(a) \(\frac{3}{m + 2n} - \frac{2}{m - 3n}\)
= \(\frac{3(m - 3n) - 2(m + 2n)}{(m + 2n)(m - 3n)}\)
= \(\frac{3m - 9n - 2m - 4n}{(m + 2n)(m - 3n)}\)
= \(\frac{m - 13n}{(m + 2n)(m - 3n)}\)
(b) Let the numbers in the digit be x and z.
Hence the number is \(10x + z\).
When it is interchanged, we have \(10z + x\).
\(10z + x = 10x + z + 9\)
\(10z - z + x - 10x = 9\)
\(9z - 9x = 9 \implies z - x = 1 ... (1)\)
\(x + z = 11 ... (2)\)
From (1), \(z = 1 + x\)
\(\therefore x + 1 + x = 11\)
\(2x + 1 = 11 \implies 2x = 11 - 1 = 10\)
\(\therefore x = 5\)
\(z = 1 + x = 1 + 5 = 6\)
\(\therefore\) The original number = 56.
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