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Data Representation and Multimedia: Question 7

Syllabus 1.1, 1.2, 1.3

Structured AS 6 marks

A USB flash drive is advertised by its manufacturer as having a capacity of 16 GB. Manufacturers state storage capacity using decimal (power-of-10) prefixes, where 1 GB = 1000 x 1000 x 1000 bytes. When the drive is plugged into a computer, the operating system instead displays storage capacity using binary (power-of-2) prefixes, where 1 GiB = 1024 x 1024 x 1024 bytes.

(a) Calculate the exact number of bytes in the manufacturer's stated capacity of 16 GB. [1]

(b) Calculate this number of bytes as a number of GiB, giving your answer to 2 decimal places. [3]

(c) Explain why the capacity shown by the operating system (in GiB) appears to be a smaller number than the capacity printed on the packaging (in GB), even though both describe the same physical amount of storage. [2]

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Worked solution

Part (a): Manufacturer’s stated capacity in bytes

The manufacturer uses the decimal prefix, where 1 GB = 1000 x 1000 x 1000 bytes = 1,000,000,000 bytes.

For 16 GB:

16 x 1,000,000,000 = 16,000,000,000 bytes

[1 mark] for the correct value 16,000,000,000 bytes.

Part (b): Converting to GiB

The operating system uses the binary prefix, where 1 GiB = 1024 x 1024 x 1024 bytes.

1024 x 1024 = 1,048,576
1,048,576 x 1024 = 1,073,741,824

So 1 GiB = 1,073,741,824 bytes.

Dividing the byte count from part (a) by this value:

16,000,000,000 / 1,073,741,824 = 14.901...

Working through the division: 1,073,741,824 x 14 = 15,032,385,536, and 16,000,000,000 - 15,032,385,536 = 967,614,464. This remainder is 967,614,464 / 1,073,741,824 = 0.9012 of a further GiB. So the total is 14 + 0.9012 = 14.9012 GiB, which rounds to 14.90 GiB (2 decimal places).

[3 marks]: [1] for using 1,073,741,824 as the divisor (the correct binary GiB size), [1] for correct division working, [1] for the correctly rounded answer 14.90 GiB.

Part (c): Why the GiB figure looks smaller

Both GB and GiB measure the same underlying quantity, bytes, but they use prefixes of different sizes. The decimal prefix “giga” used for GB means exactly 1,000,000,000, while the binary prefix “gibi” used for GiB means 1,073,741,824, which is larger.

Dividing the same fixed number of bytes (16,000,000,000) by a larger unit produces a smaller resulting number, so the drive appears to hold only “14.90 GiB” on screen even though this is exactly the same 16,000,000,000 bytes advertised as “16 GB” on the packaging. No storage capacity is actually missing; it is purely a difference in how the unit is defined.

[2 marks]: [1] for identifying that GB and GiB are different-sized units (decimal vs binary prefix) applied to the same byte count, [1] for correctly concluding that no storage is actually missing or lost.

Final answers

  • (a) 16,000,000,000 bytes
  • (b) 14.90 GiB
  • (c) GB (decimal, based on 1000) and GiB (binary, based on 1024) are different-sized units for the same byte count; dividing by the larger GiB unit gives a smaller number, so no storage is actually missing