Key Points
Key Points: Problems Based on Numbers
| Situation | Let |
|---|---|
| Two natural numbers differ by (k) | (x) and (x + k) |
| Sum of two numbers is (S) | (x) and (S − x) |
| Three consecutive numbers | (x − 1, x, x + 1) |
| Two parts of a number (N) | (x) and (N − x) |
| Two-digit number | (10x + y) |
Key Points: Problems on Ages
| Situation | Let |
|---|---|
| The present age of a person | (x) years |
| Age (k) years ago | (x - k) years |
| Age (k) years hence | (x + k) years |
Key points: Problems Based on Time and Work
| Situation | Assume |
|---|---|
| A alone takes (x) days | Rate $$= \frac{1}{x}$$ |
| B alone takes (y) days | Rate $$= \frac{1}{y}$$ |
| A & B together | $$ \frac{1}{x}$$ + $$ \frac{1}{y}$$ |
|
If A is faster than B by k days |
A → x |
| Given time in hours/minutes | Convert fully to hours or days |
Key Points: Problems Based on Distance, Speed and Time
| Situation | Formula / Setup |
|---|---|
| Basic relation | $$Speed=\frac{Distance}{Time}$$ |
| Original speed | (x) km/hr |
| Increased speed | (x + k) km/hr |
| Decreased speed | (x - k) km/hr |
| Time for distance (D) | \[\frac{D}{x}hrs\] |
| Convert minutes into hours | Convert to hours: $$\frac{\text{minutes}}{60}$$ |
Key Points: Problems Based on Geometrical Figures
| Situation | Formula / Setup |
|---|---|
| Right-angled triangle | $$(\text{Hypotenuse})^2 = (\text{Base})^2 + (\text{Height})^2$$ |
| The difference between the two sides | Let sides = x, x + k |
| Side lengths | Lengths cannot be negative |
| Rectangle | Breadth = x, Length = x + k |
| Area of a rectangle | l × b |
| Perimeter of a rectangle | 2(l + b) |
Key Points: Problems on C.P. and S.P.
| Concept | Formula |
|---|---|
| Loss | Loss = C.P. - S.P. |
| Gain | Gain = S.P. - C.P. |
| Loss % | $$\frac{\text { Loss }}{\text { C.P. }} \times 100$$ |
| Gain % | $$\frac{\text { Gain }}{\text { C.P. }} \times 100$$ |
Important Questions [7]
- A positive number is divided into two parts such that the sum of the squares of the two parts is 20. The square of the larger part is 8 times the smaller part
- A man covers a distance of 100 km, travelling with a uniform speed of x km/hr. Had the speed been 5 km/hr more it would have taken 1 hour less. Find x the original speed.
- The given table shows the distance covered and the time taken by a train moving at a uniform speed along a straight track: Distance (in m) 60, 90, y Time (in sec) 2, x, 5 The values of x and y are:
- A car covers a distance of 400 km at a certain speed. Had the speed been 12 km/h more, the time taken for the journey would have been 1 hour 40 minutes less. Find the original speed of the car.
- A bus covers a distance of 240 km at a uniform speed. Due to heavy rain its speed gets reduced by 10 km/h and as such it takes two hrs longer to cover the total distance.
- Rs. 480 is divided equally among ‘x’ children. If the number of children were 20 more, then each would have got Rs. 12 less. Find ‘x’.
- The sum of the ages of Vivek and his younger brother Amit is 47 years. The product of their ages in years is 550. Find their ages.
