Question

Solve the following word problems.
(1) A two digit number and the number with digits interchanged add up to 143. In the given number the digit in unit’s place is 3 more than the digit in the ten’s place. Find the original number.
(2) Kantabai bought $1\frac{1}{2}$ kg tea and 5 kg sugar from a shop. She paid Rs 50 as return fare for rickshaw. Total expense was Rs 700. Then she realised that by ordering online the goods can be bought with free home delivery at the same price. So next month she placed the order online for 2 kg tea and 7 kg sugar. She paid Rs 880 for that. Find the rate of sugar and tea per kg.
(3) To find number of notes that Anushka had, complete the following activity.

(4) Sum of the present ages of Manish and Savita is 31. Manish’s age 3 years ago was 4 times the age of Savita. Find their present ages.
(5) In a factory the ratio of salary of skilled and unskilled workers is 5 : 3. Total salary of one day of both of them is Rs 720. Find daily wages of skilled and unskilled workers.
(6) Places A and B are 30 km apart and they are on a st raight road. Hamid travels from A to B on bike. At the same time Joseph starts from B on bike, travels towards A. They meet each other after 20 minutes. If Joseph would have started from B at the same time but in the opposite direction (instead of towards A) Hamid would have caught him after 3 hours. Find the speed of Hamid and Joseph.

Answer 1

(1) Let the number at the unit's place be x and the digit at the ten's place be y.
The number is thus 10y + x
After interchanging the digits the number becomes 10x + y.
Given that two digit number and the number with digits interchanged add up to 143.
So, 10y + x + 10x + y = 143
$$\begin{gathered} \Rightarrow 11x+11y=143 \\ \Rightarrow x+y=13.....\left(\mathrm{I}\right) \end{gathered}$$
Also, in the given number the digit in unit’s place is 3 more than the digit in the ten’s place.
So, $x-y=3.....\left(\mathrm{II}\right)$
Adding (I) and (II) we get
$$\begin{gathered} 2x=16 \\ \Rightarrow x=8 \end{gathered}$$
Putting the value of x in (I) we get
$$\begin{gathered} 8+y=13 \\ \Rightarrow y=13-8=5 \end{gathered}$$
Thus, the number is 58.

(2) Let the rate of tea be x Rs per kg and that of sugar be y Rs per kg.
When Kantabai bought the items by going to the shop,
$$\begin{gathered} \frac{3}{2}x+5y+50=700 \\ \Rightarrow 3x+10y=1300.....\left(\mathrm{I}\right) \end{gathered}$$
When Kantabai bought the items online then
$2x+7y=880.....\left(\mathrm{II}\right)$
Multiplying (I) with 2 and (II) with 3 we get
$$\begin{gathered} 6x+20y=2600.....\left(\mathrm{III}\right) \\ 6x+21y=2640.....\left(\mathrm{IV}\right) \end{gathered}$$
(IV) $-$ (III)
$y=40$
Putting the value of y = 40 in (II)
$$\begin{gathered} 2x+7\left(40\right)=880 \\ \Rightarrow 2x=880-280=600 \\ \Rightarrow x=300 \end{gathered}$$
Thus, tea is at 300 Rs per kg and sugar is 40 Rs per kg.

(3) Disclaimer: There is error in the question given. Instead of Rs 10 notes there should be Rs 100 notes.
Let the number of notes of Rs 100 be x and that of Rs 50 be y.
$$\begin{gathered} 100x+50y=2500 \\ \Rightarrow 2x+y=50.....\left(\mathrm{I}\right) \end{gathered}$$
When number of notes is interchanged so,
$$\begin{gathered} 50x+100y=2000 \\ \Rightarrow x+2y=40.....\left(\mathrm{II}\right) \end{gathered}$$
Multiply (I) with 2
$4x+2y=100.....\left(\mathrm{III}\right)$
Subtracting (III) from (II) we get
$$\begin{gathered} 3x=60 \\ \Rightarrow x=20 \end{gathered}$$$$\begin{gathered} 3x=60 \\ \Rightarrow x=20 \end{gathered}$$
Putting the value of x in (I) we get
$y=10$
Thus, there are 20 Rs 100 notes and 10 Rs 50 notes.

(4) Let the present age of Manish be x years and that of Savita be y years.
Sum of their present ages = 31
$x+y=31.....\left(\mathrm{I}\right)$
Their age 3 years ago was
Manish's age = $x-3$
Savita's age = $y-3$
Manish’s age 3 years ago was 4 times the age of Savita.
$$\begin{gathered} x-3=4\left(y-3\right) \\ \Rightarrow x-3=4y-12 \\ \Rightarrow x-4y=-9.....\left(\mathrm{II}\right) \end{gathered}$$
(I) $-$ (II) we get
$$\begin{gathered} 5y=40 \\ \Rightarrow y=8 \end{gathered}$$
Putting the value of y in (I) we get
$$\begin{gathered} x+8=31 \\ \Rightarrow x=23 \end{gathered}$$
Thus, age of Manish is 23 years and age of Savita is 8 years.

(5) Ratio of salary of skilled to unskilled workers = 5 : 3
Let one day salary of skilled person be x and that of unskilled person be y.
Their total one day salary Rs 720
$x+y=720.....\left(\mathrm{I}\right)$
Also,
$$\begin{gathered} \frac{x}{y}=\frac{5}{3} \\ \Rightarrow 3x=5y \\ \Rightarrow 3x-5y=0.....\left(\mathrm{II}\right) \end{gathered}$$
Multiplying (I) with 3 we get
$3x+3y=2160.....\left(\mathrm{III}\right)$
(III) $-$ (II)
$$\begin{gathered} 8y=2160 \\ \Rightarrow y=270 \end{gathered}$$
Putting value of y in (I) we get
$x=450$
One day salary of skilled person Rs 450 and that of unskilled person Rs 270.

(6) Let the speed of Hamid be x km/h and that of Joseph be y km/h.
When both travel in same direction so, the distance covered by them together will be 30 km.
We know $\mathrm{Speed}=\frac{\mathrm{Distance}}{\mathrm{Time}}$
They meet each other after 20 min = $\frac{20}{60}=\frac{1}{3}$ hours
$$\begin{gathered} \frac{x}{3}+\frac{y}{3}=30 \\ \Rightarrow x+y=90.....\left(\mathrm{I}\right) \end{gathered}$$
When Joseph started from point B but moved in opposite direction.
Distance travelled by Hamid $-$ Distance travelled by Joseph = 30
$$\begin{gathered} \Rightarrow 3x-3y=30 \\ \Rightarrow x-y=10.....\left(\mathrm{II}\right) \end{gathered}$$
Adding (I) and (II) we get
$$\begin{gathered} 2x=100 \\ \Rightarrow x=50 \end{gathered}$$
Putting the value of x in (II) we get
$$\begin{gathered} 50-y=10 \\ \Rightarrow y=40 \end{gathered}$$
Thus, speed of Hamid is 50 km/h and that of Joseph is 40 km/h.