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Question:

The density of a material in SI units is 128 kg m⁻³. In certain units in which the unit of length is 25 cm and the unit of mass is 50 g, the numerical value of density of the material is?

40

410

16

640

Solution:

The correct option is D

128 kg/m³ = 128 (50 g) / (25 cm)³

= 128 (50 g) / (25 cm)³

= 128 (50 g) / (25/100 m)³

= 128 (50 g) / (1/4 m)³

= 128 (50 g) / (1/64 m³)

= 128 × 50 × 64 g/m³

= 409600 g/m³

To convert to the new units, where the unit of mass is 50 g and the unit of length is 25 cm:

409600 g/m³ = (409600 g/m³) / (50 g / 25 cm)³ = (409600/ (50 g/(25/100 m)³))
= (409600 g/m³) / (50 g / (1/4 m)³)
= 409600 g/m³ / (50 g/ (1/64 m³))
= 409600/(50*64) units
= 409600/3200 units
= 128 units

Let's verify this another way:

128 kg/m³ = 128 (1000 g) / (100 cm)³ = 128000 g/ 1000000 cm³ = 0.128 g/cm³

In the new units, 1 unit of mass = 50 g and 1 unit of length = 25 cm

Therefore, 1 unit of volume = (25 cm)³ = 15625 cm³

Density in new units = (0.128 g/cm³) * (50 g / 1 unit mass) * (1 unit volume / 15625 cm³)
= (0.128 * 50 / 15625) units = 0.0004096 units

There's a discrepancy in the calculation. Let's re-examine the approach.

We have 128 kg/m³. Let's convert this to the new units:

1 kg = 1000 g = 20 units of mass (since 1 unit of mass = 50 g)
1 m = 100 cm = 4 units of length (since 1 unit of length = 25 cm)
1 m³ = (4 units of length)³ = 64 units of volume

So, 128 kg/m³ = 128 * 20 units of mass / 64 units of volume = 40 units of density