SOLUTION: thank you for helping math Allegation Problem: Word Problem In what proportion must plain coal tar be combined with a 2% coal tar ointment to make a 4% coal tar ointment? t

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Question 815060: thank you for helping math
Allegation Problem: Word Problem
In what proportion must plain coal tar be combined with a 2% coal tar ointment to make a 4% coal tar ointment?
thank you again.

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
2% is +2%2F100+=+.02+
4% is +4%2F100+=+.04+
Plain coal tar is 100% = +100%2F100+=+1+
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Let +a = units of 2% coal tar needed
Let +b+ = units of 100% coal tar needed
( I will choose units for a and b
so that a+%2B+b+=+10+ units
---------------------------------------------
(1) +%28+.02a+%2B+1%2Ab+%29+%2F+%28+a+%2B+b+%29+=+.04+
(2) +a+%2B+b+=+10+
---------------
(1) +.02a+%2B+b+=+.04%2A%28+a+%2B+b+%29+
(1) +.02a+%2B+b+=+.04a+%2B+.04b+%29+
(1) +.96b+-+.02a+=+0+
and
(2) +a+=+10+-+b+
by substitution:
(1) +.96b+-.02%2A%28+10+-+b+%29+=+0+%29+
(1) +.96b+-+.2+%2B+.02b+=+0+
(1) +.98b+=+.2+
(1) +b+=+10%2F49+
and, since
(2) +a+=+10+-+b+
(2) +a+=+10+-+10%2F49+
(2) +a+=+490%2F49+-+10%2F49+
(2) +a+=+480%2F49+
------------------
The proportion of plain coal tar to 2% coal tar
should be:
+b%2Fa+=+%28+1%2F49+%29+%2F+%28+480%2F49+%29+
+b%2Fa+=+1%2F480++
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check answer:
(1) +%28+.02a+%2B+1%2Ab+%29+%2F+10+=+.04+
(1) ++.02a+%2B+b++=+.4+
(1) +.02a+=+.4+-+b+
(1) +.02%2A%28+480%2F49+%29+=+.4+-+10%2F49+
Multiply both sides by +49+
(1) +.02%2A480+=+.4%2A49+-+10+
(1) +9.6+=+19.6+-+10+
(1) +9.6+=+9.6+
OK