(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...
Question 1 Report
(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.
Answer Details
(a) Common denominator \((m+2n)(m-3n)\):
\[\frac{3}{m+2n} - \frac{2}{m-3n} = \frac{3(m-3n) - 2(m+2n)}{(m+2n)(m-3n)}\]
\[= \frac{3m - 9n - 2m - 4n}{(m+2n)(m-3n)} = \frac{m - 13n}{(m+2n)(m-3n)}\]
(b) Let the tens digit be \(t\) and the units digit be \(u\). The number is \(10t + u\).
Sum of digits: \(t + u = 11\).
Interchanging the digits gives \(10u + t\), which exceeds the original by 9:
\[10u + t = (10t + u) + 9 \;\Rightarrow\; 9u - 9t = 9 \;\Rightarrow\; u - t = 1\]
Solving \(t + u = 11\) and \(u - t = 1\): \(u = 6,\; t = 5\).
The number is \(\mathbf{56}\).
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