# GATE (TF) Textile 2020 Question Paper Solution | GATE/2020/TF/46

###### Question 46 (Textile Engineering & Fibre Science)

Two types of polyester staple fibres of fineness 3 and 6 denier and having the same length are mixed in a ratio of 2:3 by weight. The mean fibre fineness (denier) of the mix (rounded off to 2 decimal places) is __4.25 to 4.35__.

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Given in the question:

Fineness of first polyester fibre=3 denier

Fineness of second polyester fibre=6 denier

Both have same length in the ratio 2:3 by weight

The mean fibre fineness of the mix=?

Calculation:

Let us assume the total weight of the mix(w) =100 gm

Then,
Weight of the first polyester fibre in the mix (w1)=100 x (2/5)
w1=40 gm

Weight of the second polyester fibre in the mix (w2)=100 x (3/5)
w2 =60 gm

Here, L1=L2

Where , L1=Length of first polyester fibre
and L2=Length of second polyester fibre

By definition of denier– Denier=weight in gm/9000 meter length

From first polyester fibre:

Weight of 9000 meter Length fibre=3 gm

1 gm weight have=9000/3

Then,

40 gm weight have=9000 x 40/3
L1=120000 mtr

From second polyester fibre:

Weight of 9000 meter Length fibre=6 gm

1 gm weight have=9000/6

Then,

60 gm weight have=9000 x 60/6
L2=90000 mtr

Total length of mix (L)=L1+L2

L=120000+90000

L=210000 mtr

By definition of denier

We have,
Weight 210000 mtr length of polyester =100 gm
Then ,
Weight of 1 mtr length of polyester=100/210000

Weight of 9000 mtr length of polyester=100 x 9000/210000

Fineness of the mix=4.28 denier (Answer)

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