A recipe for oatmeal raisin cookies calls for #1 2/3# cups of flour to make 4 dozen cookies. How many cups of flour are needed to make 6 dozen cookies?

2 Answers
CountryGirl
Apr 13, 2018

#3 1/3# cups

Explanation:

The key to this question is to find how much flour the recipe needs per dozen, and the you can calculate how much is needed for #6# dozen (or any amount that you like).

So we know that:

#"3 dozen" = 1 2/3#

#3d = 1 2/3#

So to figure out #1# dozen, just divide both sides by #3# to isolate the variable #d#. The easiest way to divide a mixed number is to first turn it into an improper fraction and then multiply it by the reciprocal of #3#.

The easiest way to understand that rather confusing sentence is for me to show you:

#3d = 1 2/3#

#3d = 5/3# (so I just turned it into an improper fraction)

#d = 5/3 -: 3# (now I divided both sides by 3)

#d = 5/3 -: 3/1# (remember that anything divided by 1 is itself - so this just turns 3 into a fraction without changing the value)

#d = 5/3 xx 1/3# (now it's being multiplied by the reciprocal - or flipped version - of 3 because that's how fractions are divided )

#d = (5xx1)/(3xx3) rarr 5/9 #

So each dozen needs #5/9# of a cup. But what is #6# dozen?

#6d = ?#

#6(5/9) = ?#

#6/1 xx 5/9 = ?# (remember to turn it into a fraction!)

#stackrel(color(red)2)cancel6/1 xx 5/stackrel(color(red)3)cancel9# (cross simplify )

#2/1 xx 5/3 rarr (2 xx 5)/(1 xx 3) rarr 10/3#

Now we'll turn #10/3# back into a mixed number:

#10/3 rarr 3 1/3# cups.

Ben
Apr 13, 2018

2.5 cups of flour are needed for 6 dozen cookies.

Explanation:

This solution is found by calculating a ratio between the two volumes of cookies. If you want to increase the volume of cookies to 6 from 4, the ratio would be 4:6.

This ratio means "for every 4 of one thing, there will be 6 of the new thing in the scaled ratio" This can be interpreted into a scaling fraction, #6//4#, then simplified to #3//2#

Now that we have a fraction for scaling, it's easy to determine the increase in flour required:

#(1 2/3)cancel("flour for 4 dozen") xx3/2"flour for 6 dozen"/cancel("flour for 4 dozen")=x" flour for 6 dozen"#

#(1+2/3)xx3/2=x#

#3/2+(cancel(2/3*3/2)color(red)(rarr1))=x#

#color(green)(1 3/2=2.5=x)#

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