EXPLANATION
Let's see the facts:
Banana Composition:
0.7 grams of protein
0.2 grams of fat
13.5 gram of carbohydrates
Mandarin composition:
0.5 gram of protein
0.3 grams of fat
8.5 grams of carbohydrates
Grapes composition:
0.3 grams of protein
0.2 grams of fat
8 grams of carbohydrates
Needed---->
3.2 grams of protein
1.5 grams of fat
60.5 gram of cabohydrates
Let x be the servings of bananas, y the servings of mandarin and z the serving of grapes, we will have a system of equation with the following form:
(1) 0.7x + 0.5y + 0.3z = 3.2
(2) 0.2x + 0.3y + 0.2z = 1.5
(3) 13.5x + 8.5y + 8z = 60.5
Now, we must solve for x,y and z;
Isolate x for 0.7x + 0.5y + 0.3z = 3.2:
Multiply both sides by 10:
0.7x*10+0.5y*10+0.3z*10=3.2*10
Refine:
7x + 5y + 3z = 32
Subtract 5y + 3z from both sides:
7x + 5y + 3z - (5y + 3z) = 32 - (5y+3z)
Simplify:
7x = 32 - 5y -3z
Divide both sides by 7:
7x/7 = 32/7 - 5y/7 -3z/7
Simplify:
x= (32-5y-3z)/7
Substitute x= (32-5y-3z)/7
0.2[(32-5y-3z)/7] + 0.3y + 0.2z = 1.5
13.5[(32-5y-3z)/7] + 8.5y + 8z = 60.5
Simplify 0.2[(32-5y-3z)/7] + 0.3y + 0.2z = 1.5:
Multiply the fractions: a*b/c = (a*b)/c
0.2[(32-5y-3z)/7] = 0.02857* (32-5y-3z)
Expand 0.02857* (32-5y-3z):
Distribute parentheses:
=0.02857*32 -0.02857*5y -0.02857*3z
Multiply the numbers:
=0.91428 - 0.14285y - 0.08571z + 0.3y + 0.2z = 1.5
Add terms and simplify:
0.15714y + 0.11428z + 0.91428 = 1.5
Let's continue with the third equation:
13.5[(32-5y-3z)/7] + 8.5y + 8z = 60.5
Distribute terms:
1.92857*(32-5y-3z) + 8.5y + 8z = 60.5
Distribute parentheses:
1.92857*32 -5*1.92857y -1.92857*3z + 8.5y + 8z = 60.5
Simplify:
61.71428 -9.64285y -5.78571z + 8.5y + 8z = 60.5
Adding similar terms:
-1.14285y +2.21428z +61.71428 = 60.5
Our system of equations is now:
0.15714y+0.11428z+0.91428s =1.5
-1.14285y+2.21428z+61.71428 =60.5
Isolate y for 0.15714y+0.11428z+0.91428s =1.5:
Subtract 0.11428 from both sides:
0.15714y+0.11428z+0.91428 -0.11428z=1.5-0.11428z
Simplify:
0.15714y+0.91428=1.5-0.11428z
Subtract 0.91428 from both sides:
0.15714y+0.91428-0.91428 =1.5-0.11428z-0.91428
Simplify:
0.15714y=-0.11428z+0.58571
Divide both sides by 0.15714:
0.15714/0.15714y=-0.11428/0.15714z+0.58571/0.15714
Simplify:
y=(-0.11428z+0.58571)/0.15714
Substitute y=(-0.11428z+0.58571)/0.15714:
-1.14285*[(-0.11428z+0.58571)/0.15714]+2.21428z+61.71428=60.5
Simplify:
3.04545z+57.45454=60.5
Isolate z for 3.04545z+57.45454=60.5:
Subtract 57.45454 from both sides
3.04545z+57.45454-57.45454 =60.5-57.45454
Simplify:
3.04545z=3.04545
Divide both sides by 3.04545:
Simplify:
z=1
For y=(-0.11428z+0.58571)/0.15714
Substitute z=1
y=(-0.11428*1+0.58571)/0.15714
Simplifying:
y=3
For x= (32-5y-3z)/7 substitute y=3 and z=1
x= (32-5*3-3*1)/7
Simplifying:
x=2
The solutions to the systems of equations are:
x=2, y=3 and z=1
So, in order to supply exactly 3.2 grams of protein, 1.5 grams of fat and 60.5 grams of carbohydrates, we need 2 servings of banana, 3 servings of mandarin and 1 serving of grapes.