Lauren Gonzalez

Question

A sandwich is ordering apples and grapes to make chicken salad. Apples cost(blank) per pound and grapes cost $2.60 per pound. If they ordered a total of 20 pound of apples and grapes and paid $35.80, how many pound of grapes did they order?

Answer 1

Less than 13 pounds of grapes, as 14 x $2.60 exceeds $35.80, You could order as low as zero pounds of grapes, then the price of apple pounds would be $35.80/20 =$1.79 per pound

This problem has an indeterminate number of solutions. There must be more information to get a unique solution. But best answers that give an exact or integer number of cents are 80 cents,98 cents $1.25 or $1.79 per pound of apples, for 11, 10, 8 or 0 pounds of grapes

Let G=pounds of grapes ordered. 0<G<13

Let Pa = price of apples per pound and Pg=price of grapes=2.60

Let A=pounds of apples ordered, and G=pounds of grapes ordered

A+G=20 A=20-G

PaA+PgG=35.80

Pa(20-G)+2.60G=35.80

35.80/2.60 = 13.77 pounds of grapes, but no money is left to buy apples, unless Pa=0

G<13.77

You have 3 unknowns, Pa, A and G

and only 2 equations: A+G=20 and paA+2.70G=35.80

There is an indeterminate number of solutions

You need more information, a 3rd equation to get a unique answer

If G=13 then PgG= 2.60(13)= 33.80 leaving $2 left to buy apples. Then Pg=$2/7=14+ cents per apple pound

The only way there might be a unique solution is if Pg is a whole number of cents, A and B are integers, so there are not that many numbers to check

Let G=12 then PgG=2.60x12=31.20 leaving $4.60 for 8 apples, then Pa=4.60/8=57+ cents

Let G=11 then PgG=2.60x11=28.60 leaving $7.20 for 9 apples. Then Pa=7.20/9= exactly 80 cents per apple pound

If all other possibilities are fractions of cents, then $.80 per apple pound is the unique solution

Let G=10 then PgG=$26, leaving $9.80 for 10 apples or Pa=9.80/10 = exactly 98 cents per apple pounds

So, there is no unique solution. 80 cents and 98 cents per apple are possible, both giving an integer number of cents

Let G=9 then PgG=$23.40 leaving $12.40 for 11 apples. 11 does not divide 12.40 evenly

Let G=8, then PgG=$20.80 leaving $15 for 12 apples. Pa=15/12=5/4=$1.25 per pound of apples

Now we have 3 possible solutions, Pg = 80 cents, 98 cents or $1.25 per pound of apples

Let G=7 then PgG = $18.20 leaving $17.80 for 13 apple pounds. 13 does not divide 17.8 evenly

Let G=6, then PgG=$15.60 leaving $20.20 for 14 apple pounds. 20.20/14=10.10/7 which is not an integer for cents

Let G=5, then PgG=$13.00 leaving $12.80 for 15 pounds of apples. 12.80/15 is not an integer

Let G=4 then PgG= $10,.40 leaving $25.40 for 16 pounds of apples 25.40/16=6.35/4 which is not an integer

Let G=3 thenPgG =$7.80 leaving $28 for 17 pounds 28/17 is not an integer

Let G=2 then PgG = $5.20 leaving $30.60 for 18 pounds 30.6/18=15.3/9=5.1/3 is not an integer

Let G=1 then PgG = $2.60 leaving $33.20 for 19 pounds. 33.2/19 = $1.69+ is not an integer for cents

Let G=0, then PgG=0 leaving all $35.80 for 20 pounds of apples $35.80/20= $1.79

Best answers may be 80 cents, 98 cents, $1.25 or $1.79 per pound of apples. Or if you allowed fractions for the cents per pound, the price of apples could be anywhere from zero to $1.79, corresponding to 11 to zero pounds of grapes ordered