Scaling is an important topic in Quantitative Aptitude and is frequently tested in competitive exams such as GATE, CAT, and other aptitude-based tests. The concept of scaling is based on ratio and proportion and helps us understand how different physical quantities change when the size of an object is increased or decreased.
Scaling refers to the process of enlarging or reducing the size of any object by certaim factor while maintaining its shape. This factor is known as the scalar factor.
If an object is scaled by a factor k, all its dimensions change proportionally according to specific mathematical rules.
Scaling Rules For Different Quantities
When an object is scaled by a factor k:
- Length sacles as k
- Perimeter scales as k
- Area scales as k
- Volume scales as k3
These rules form the foundation of most scaling problems in aptitude.
Length Scaling
If the length of an object is increased or decreased by a scale factor k, the new length becomes:
New Length = k x Original Length
Example: If the side of a square is doubled, its new side becomes twice the original length.
Area Scaling
Area depends on two dimensions, so it changes as the square of the scale factor.
New Area = k2 × Original Area
If the radius of a circle becomes 3 times, its area becomes 32=93^2 = 932=9 times the original area.
Volume Scaling
Volume depends on three dimensions, so it changes as the cube of the scale factor.
New Volume = k3 × Original Volume
Example: If the edge of a cube is doubled, its volume increases by 23= 8 times.
Percentage Change Using Scaling
Scaling is often combined with percentage increase or decrease.
Example: If the radius of a sphere is increased by 20%, New radius = 1.2r
Since volume ∝ r3:
New Volume = (1.2)3=1.728
So, the volume increases by 72.8%.
Scaling in Speed, Time, and Distance
Scaling also appears in problems involving speed and time.
- If distance is scaled and speed remains constant, time scales proportionally.
- If speed is scaled and distance remain constant , time scales inversly.
Time ∝ Speed /Distance
Solved Questions on Scaling
Question 1: The side of a square is increased by 50%. Find the percentage increase in its area.
Solution:
New Side = 1.5s
Area ∝ side 2
New area = (1.5)2 = 2.25Increase = 125%
Question 2: The radius of a sphere is increased by 10%.Find the percentage increase in volume.
Solution :
New radius = 1.1r
Volume ∝ r3
New Volume = (1.1)3 = 1.331
Increase = 0.331×100 = 33.1% Increase
Question 3: The length, breadth, and height of a cuboid are doubled.How many times does its volume increase?
Solution:
Volume ∝ l × b × h
New Volume=2 × 2 × 2 = 8
Volume increases 8 times
Question 4: The base of a triangle is increased by 25%, while its height is decreased by 20%. Find the net percentage change in area.
Solution:
Area ∝ base × height
New Area = 1.25 × 0.8 = 1
No change in area
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Unsolved Question on Scaling
Question 1:The side of a square is increased by 30%.Find the percentage increase in its area.
Question 2: The radius of a circle is reduced by 20%. By what percentage does its area decrease?
Question 3:The edge of a cube is increased in the ratio 3 : 5. Find the ratio of the volumes of the original and new cube.
Question 4:The radius of a cylinder is doubled while its height is reduced to half.Find the ratio of the new volume to the original volume.