Question

There are two conversions that we have to make in the below problems:
Convert into minutes and seconds:
(a) 75 seconds
(b) 285 seconds
(c) 781 seconds
Convert into years and months:
(d) 95 months
(e) 124 months
(f) 45 months

Answer 1

Hint: We have to make two kinds of conversions, one is in minutes and seconds and another one is into years and months. We know the relationship between minutes and seconds i.e. $1\text{minute}=60\text{seconds}$ Using this relation we can convert the seconds given in the question into minutes. We also know the relationship between years and months which is given by the following formula $1\text{year}=12\text{months}$. Using this relation we can convert the months into years and months.

Complete step by step answer:
First of all we are doing the first conversion i.e. seconds to minutes which we are going to do by using the following formula:
$1\text{minute}=60\text{seconds}$…….. Eq. (1)
Solving the option (a) 75 seconds we get,
The above formula in minutes and seconds is equal to:
 $1\text{minute}=60\text{seconds}$
Dividing 60 on both the sides will give us how many minutes in one second.
$\begin{align}
  & \dfrac{1}{60}\text{minute}=\dfrac{60}{60}\text{seconds} \\
 & \Rightarrow \dfrac{1}{60}\text{minute}=1\text{second}.........\text{Eq}\text{.(2)} \\
\end{align}$
Now, multiplying 75 on both the sides we get,
$\dfrac{75}{60}\text{minutes}=75\text{seconds}$
Dividing numerator and denominator of the left hand side by 15 of the above equation we get,
$\Rightarrow \dfrac{5}{4}\text{minutes}=75\text{seconds}$
We can write $\dfrac{5}{4}$ as $1+\dfrac{1}{4}$ . This form is written by writing 5 as $Divisor\times Quotient+\text{Remainder}$ in $\dfrac{5}{4}$
In the above expression divisor is 4, quotient is what we get after dividing 5 by 4 which is 1 and remainder is also 1 so substituting these values in the above equation we get,
$\begin{align}
  & \dfrac{4\times 1+1}{4} \\
 & =1+\dfrac{1}{4} \\
\end{align}$
So, using this relation in the above problem we get,
\[\begin{align}
  & \left( 1+\dfrac{1}{4} \right)\text{minutes}=75\text{seconds} \\
 & \Rightarrow \text{1minute and }\dfrac{1}{4}\text{minute}=75\text{seconds} \\
\end{align}\]
We can find \[\dfrac{1}{4}\text{minute}\] in seconds by multiplying it by 60.
$\dfrac{1}{4}\left( 60 \right)=15\text{seconds}$
Hence, 75 seconds is equal to 1 minute and 15 seconds.
Solving option (b) 285 seconds we get,
The relation between minutes and seconds that we derived in the previous part is:
$\dfrac{1}{60}\text{minute}=1\text{second}$
Multiplying 285 on both the sides we get,
$\dfrac{1}{60}\left( 285 \right)\text{minutes}=\left( 285 \right)\text{seconds}$
Dividing numerator and denominator of the left hand side by 15 of the above equation we get,
$\dfrac{1}{4}\left( 19 \right)\text{minutes}=\left( 285 \right)\text{seconds}$
We can write $\dfrac{19}{4}$ as $4+\dfrac{3}{4}$ so using this relation in the above problem we get,
$\begin{align}
  & \left( 4+\dfrac{3}{4} \right)\text{minutes}=\left( 285 \right)\text{seconds} \\
 & \Rightarrow \text{4minutes and }\dfrac{3}{4}\text{minutes}=\left( 285 \right)\text{seconds } \\
\end{align}$
We can find \[\dfrac{3}{4}\text{minute}\] in seconds by multiplying it by 60.
$\begin{align}
  & \dfrac{3}{4}\times 60 \\
 & =3\left( 15 \right)=45 \\
\end{align}$
Hence, 285 seconds is equal to 4 minutes and 45 seconds.
Solving option (c) 781 seconds we get,
The relation between minutes and seconds that we derived in the previous part is:
$\dfrac{1}{60}\text{minute}=1\text{second}$
Multiplying 781 on both the sides we get,
$\dfrac{1}{60}\left( 781 \right)\text{minutes}=\left( 781 \right)\text{seconds}$
We can write $\dfrac{781}{60}$ as $13+\dfrac{1}{60}$ and using this form in the above equation we get,
$\begin{align}
  & \left( 13+\dfrac{1}{60} \right)\text{minutes}=\left( 781 \right)\text{seconds} \\
 & \Rightarrow 13\text{minutes and }\dfrac{1}{60}\text{minutes}=\left( 781 \right)\text{seconds} \\
\end{align}$
We can convert $\dfrac{1}{60}\text{minutes}$ into seconds by multiplying it by 60.
$\dfrac{1}{60}\times 60=1\text{second}$
Hence, we have converted 781 seconds into 13minutes and 1 second.
Now, we are doing the second conversion of months and years as follows:
$1\text{year}=12\text{months}$
Solving option (d) 95 months using above relation between months and year we get,
$1\text{year}=12\text{months}$
Dividing 12 on both the sides so that we will get how many years does a 1 month contain.
$\begin{align}
  & \dfrac{1}{12}\text{year}=\dfrac{12}{12}\text{months} \\
 & \Rightarrow \dfrac{1}{12}\text{year}=1\text{month} \\
\end{align}$
Multiplying 95 on both the sides we get,
$\dfrac{95}{12}\text{years}=95\text{months}$
We want answer in months and years so we can write $\dfrac{95}{12}$ as $7+\dfrac{11}{12}$ in the above equation we get,
$\begin{align}
  & \left( 7+\dfrac{11}{12} \right)\text{years}=95\text{months} \\
 & \Rightarrow \text{7years and }\dfrac{11}{12}\text{year}=95\text{months} \\
\end{align}$
We can convert $\dfrac{11}{12}\text{year}$ into months by multiplying it by 12.
$\dfrac{11}{12}\times 12=11months$
Hence, we have converted 95 months into 7 years and 11 months.
Solving option (e) 124 months we get,
$1\text{year}=12\text{months}$
Dividing 12 on both the sides so that we will get how many years does a 1 month contain.
$\begin{align}
  & \dfrac{1}{12}\text{year}=\dfrac{12}{12}\text{months} \\
 & \Rightarrow \dfrac{1}{12}\text{year}=1\text{month} \\
\end{align}$
Multiplying 124 on both the sides we get,
$\begin{align}
  & \dfrac{124}{12}\text{years}=124\text{months} \\
 & \Rightarrow \dfrac{31}{3}\text{years}=124\text{months} \\
\end{align}$
We want answer in months and years so we can write $\dfrac{31}{3}$ as $10+\dfrac{1}{3}$ in the above equation we get,
$\begin{align}
  & \left( 10+\dfrac{1}{3} \right)\text{years}=124\text{months} \\
 & \Rightarrow 10\text{years and }\dfrac{1}{3}year=124\text{months } \\
\end{align}$
We can convert $\dfrac{1}{3}\text{year}$ into months by multiplying it by 12.
$\dfrac{1}{3}\times 12=4months$
Hence, we have converted 124 months into 10 years and 4 months.
Solving the option (f) 45 months we get,
$1\text{year}=12\text{months}$
Dividing 12 on both the sides so that we will get how many years does a 1 month contain.
$\begin{align}
  & \dfrac{1}{12}\text{year}=\dfrac{12}{12}\text{months} \\
 & \Rightarrow \dfrac{1}{12}\text{year}=1\text{month} \\
\end{align}$
Multiplying 45 on both the sides we get,
$\begin{align}
  & \dfrac{45}{12}\text{years}=45\text{months} \\
 & \Rightarrow \dfrac{15}{4}\text{years}=45\text{months} \\
\end{align}$
We want answer in months and years so we can write $\dfrac{15}{4}$ as $3+\dfrac{3}{4}$ in the above equation we get,
$\begin{align}
  & \left( 3+\dfrac{3}{4} \right)\text{years}=45\text{months} \\
 & \Rightarrow 3\text{years and }\dfrac{3}{4}year=45\text{months } \\
\end{align}$
We can convert $\dfrac{3}{4}\text{year}$ into months by multiplying it by 12.
$\dfrac{3}{4}\times 12=9months$
Hence, we have converted 45 months into 3 years and 9 months.

Note: In the above conversions we have converted the fraction part of minutes and year into seconds and months. If you won’t convert the fraction part of minutes and year into seconds and months then also the answer is correct.
For example, in option (f) 45 months,
We have got $\dfrac{15}{4}years$ then we have converted these years into years and months as 3 years and 9 months instead of converting them into years and months you can write answers only in years as 3.75 years.