Discussion

Explanation:

Number of pencils is 30% of total items = 3/10 × total items
Now, ‘3/10 × total items’ should  be an integer, hence total items should be a multiple of 10.
Also, 40% of these are sold = 2/5 × 3/10 × total items = 3/25 × total items.
Now, ‘3/25 × total items’ should be an integer, hence total items should also be a multiple of 25.
⇒ Total items should be a multiple of LCM (10, 25) = 50.
Let the total number of items initially be ‘50i’. [i in an integer]

Now, 30% of erasers are sold = 3/10 × erasers
Now, ‘3/10 × erasers’ should be an integer, hence number of erasers should be a multiple of 10.
Let the number of erasers be ‘10e’. [e in an integer]

Now, items sold  are
Pencils = 40% of 30% of 50i = 6i
Sharpeners = 160
30% of Erasers = 3e

Total 40% of items are sold
⇒ 40% of 50i = 6i + 160 + 3e
⇒ 20i = 6i + 160 + 3e
⇒ 14i = 160 + 3e

⇒ i = $\frac{160+3\mathrm{e}}{14}$

⇒ i = $\frac{154+4+3\mathrm{e}}{14}$

⇒ i = 11 + (4 + 3e)/14

Since, i & e are both integers, possible values of a should be such that (4 + 3e) should  be divisible by 14.
∴ least possible value of ‘e’ = 8

⇒ Least possible value of i = 11 + 2 = 13
⇒ Least possible value of items = 50 × 13 = 650

Hence, 650

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