Suppose $A$ and $B$ can complete a work together in 10 days. If $B$ alone can complete the work in 15 days, then in how many days can $A$ alone finish the work?<\/p>\n
[amp_mcq option1=”20 days” option2=”24 days” option3=”25 days” option4=”30 days” correct=”option4″]<\/p>\n
Work done = Rate \u00d7 Time.
\nRate = Work \/ Time.<\/p>\n
A and B together complete the work in 10 days.
\nTheir combined rate is $R_A + R_B$.
\n$(R_A + R_B) \\times 10 = W$
\n$R_A + R_B = \\frac{W}{10}$<\/p>\n
B alone completes the work in 15 days.
\nB’s rate is $R_B$.
\n$R_B \\times 15 = W$
\n$R_B = \\frac{W}{15}$<\/p>\n
We want to find the time it takes for A alone to finish the work. Let this time be $T_A$.
\n$R_A \\times T_A = W$
\n$T_A = \\frac{W}{R_A}$<\/p>\n
Substitute the value of $R_B$ into the combined rate equation:
\n$R_A + \\frac{W}{15} = \\frac{W}{10}$<\/p>\n
Solve for $R_A$:
\n$R_A = \\frac{W}{10} – \\frac{W}{15}$<\/p>\n
Find a common denominator for the fractions (LCM of 10 and 15 is 30):
\n$R_A = \\frac{3W}{30} – \\frac{2W}{30}$
\n$R_A = \\frac{3W – 2W}{30} = \\frac{W}{30}$<\/p>\n
Now, calculate the time taken for A alone:
\n$T_A = \\frac{W}{R_A} = \\frac{W}{\\frac{W}{30}}$
\n$T_A = W \\times \\frac{30}{W} = 30$<\/p>\n