 ### writing equations, expressions, statements -- mixtures     | Intro | More Of | Off | Mixing Stuff | Your Turn |

Intro
More Of
The phrase that "collection of Jack's of marbles and things from his grandfather" refers to Jack's inherited marble collection. The prepositional phrases each describe in more detail the collection.

"The result of the addition of \$ 12 and the cost of a new part" also include propositional phrases and may, with other examples, be considered as follows.

1.
 result of the addition of \$ 12 and the cost of a new part (result) (12) + (cost) 12 + x

2.
 6% sales tax on the result of the addition of twelve dollars and the cost of a new part (6%)(result) (.06)(12 + (cost)) .06(12 + x)

Prepositional phrases provide additional detail. Consider these examples.
3.
 half of a number (number)/2 x/2 or .5x

4.
 a quarter of a half of a number (a quarter) ( )/4 (of half of a number)/4 ( (number)/2 )/4 (number)/8 x/8

5.
 8% annual interest paid monthly (8%)(interest per month) .08(interest)/12

Off & Of
Most people have heard the expression "A glass which is half empty is also half full."

So, 50% OFF something equals 50% OF something.

Similarly 60% OFF something equals 40% OF something.

Similarly 25% OFF something equals 75% OF something.

This full-empty, on-off, relationship is very useful algebraically. See the next examples
6.
 A jet skii is sold for 30% off and the sale price is \$4000. What was the original price? (40% OFF the original price) = (sale price) (60% OF the original price) = (sale price) (.60)(original price) = 4000 .60x = 4000

7.  Mixing Stuff

The usual story line of a mixture problem goes something like:

Some strong stuff is added to weak stuff to make a usable mixture.

The usual pictures of mixtures look like these.   8. Fifty bottles of 70% blue liquid and a number of bottles of 30% blue liquid are mixed in a big container to obtain a new mixture contains 55% blue liquid. How many bottles of 30% blue liquid are required?   mix some 30% stuff with 50 units of 70% stuff to get 55% stuff 30%(stuff) +70%(stuff) = 55%(stuff) .30(x) +.70(50) = .55(x + 50)

9. Some 20% pure stuff is in a pot. To this is added 30 units of 80% pure stuff to fill the pot with 45% pure stuff.   (20% stuff) + (80% stuff) = (45% stuff) .2(x) + .8(30) = .45(x + 30)

10. Some candy costing \$1.85 per unit is mixed with some other candy costing \$2.45 per unit to make 24 units of \$2 per unit candy. (\$1.85 candy) + (\$2.45 candy) = (\$2.00 candy) 1.85(x) + 2.45(24-x) = 2(24)

 Some 70% punch is mixed with some 40% punch to produce 2 gallons of 62% punch. Answer (70% punch) + (40% punch) = (62% punch) Answer (70%(some punch) + (40%)(rest of the punch) = (62%)(2 gallons punch) Answer .70(x) + .40(2 - x) = .62(2)
 There are 40 nickels and dimes in a bank which holds \$2.65. How many of each coin are there? Answer (nickels) + (dimes) = \$2.65 Answer .05(nickels) + .10(dimes) = 2.65 Answer .05(x) + .10(40 - x) = 2.65 Answer 5(x) + 10(40 - x) = 265 Answer 5x + 400 - 10x = 265 Answer 400 - 5x = 265 Answer 400 - 400 - 5x = 265 - 400 Answer - 5x = - 135 Answer 5x = 135 Answer 5x/5 = 135/5 Answer x = 27 Answer 27 nickels, 13 dimes              © Feb. 14, 2003 www.mathnstuff.com/math/algebra/qb/awrite4.htm