Scaling

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 = 2.25

Increase = 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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• Grouping
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• Cutting

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.