Word problems, help!!

Question

When the product of three consecutive integers is decreased by the cube of the smallest, the result is 85. What are the integers?

 

Three dozen eggs and five loaves of bread cost $6.90; four dozen eggs and two loaves of bread cost $5.70. Find the prices for a dozen eggs and for a loaf of bread.

 

 

A chemist prepared 100 mL of 15% saline solution by using a 20% solution, an 18% solution and 40 mL of a 10% solution. How much of the 20% and 18% solutions were used?

Answer 1

Dear George,

 

Star by highlighting or underlining every key word or number in the sentence.

 

                 X* (X+1)*(X+2)   -  X^(³)                                                                                              =85

When the product of three consecutive integers is decreased by the cube of the smallest, the result is 85.

 

 

Question:  What are the integers?  

 

 

You need to solve for "X," so  you can substitute the value and find each consecutive integer.

 

 

Let "X=the first consecutive integer and the smallest

 

Then X+1= 2nd consecutive integer

 

and  X+2=3rd consecutive integer Note: "Product" means to multiply.

 

 

X* (X+1)*(X+2) - X^(³) =85     Simplify by multiplying (X+1)(X+2).

 

X (X² +1X +2X+2) - x³ = 85       Collect terms.

 

X(X² +3X+2)- X³ =85                 Now, use the distributive property.

 

X³ +3X² +2X - X³ =85               The X cubes cancel each other out.  Subtract 85 from both sides of the equation.

 

3X² +2X -85=0           You can now use the Quadratic Formula to find the X-intercepts or solutions to the equation.

                               

                                   Note:  This is where "Y=0."

 

a=3, b=2, c=-85                  

 

X= [-b  +/- √(b² - 4(a)(c)) ]/2a            Just substitute your values into the equation, and you can find "X," the

 

first consecutive number,   (X+1), the second consecutive integer, and (X+2), the third consecutive integer. 

 

Remember that you are not looking for a mixed number nor an improper fraction for the solution of "X"  but a whole

 

number, an integer.

 

I am going to let you finish your problem; I hope that I was helpful; please give me a thumbs up if I was.

 

                                                                      Susan C.

 

Answer 2

x*(x+1)*(x+2)-x^3=85

x^3+3x^2+2x-x^3=85

3x^2+2x-85=0

factor

(3x+17)(x-5)=0

x-5=0

x=5

x+1=6

x+2=7

 

3x+5y=6.9

4x+2y=5.7

 

12x+20y=27.6

12x+6y=17.1

14y=10.5

y=$0.75 for a loaf of bread

3x+5(0.75)=6.9

3x+3.75=6.9

3x=6.9-3.75

3x=3.15

x=$1.05 for a dozen eggs

 

100(15%)=40(10%)+x(20%)+y(18%)

x+y=60  (100-40=60)

x+y=60 ----> x=60-y

100(15%)=40(10%)+(60-y)(20%)+y(18%)

100(0.15)=40(0.1)+(60-y)(0.2)+y(0.18)

15=4+12-0.2y+0.18y

15=16-0.02y

0.02y=16-15

0.02y=1

2y=100

y=50 L of the 18% solution

x=60-50=10 L of the 20% solution

Answer 3

The product of three consecutive integers can be represented as three separate variables if you wish.

Let's say m*n*o.

But the fact that they are consecutive means n = m+1, and o = n + 1 or o = m+2.

We'd be able to solve a single equation in terms of a single unknown, so let's use those facts to write the product of the three integers in terms of m.

 

Go ahead and do that, now.

 

 

 

 

 

 

 

 

 

 

 

 

 

You should have got, after simplifying, m^3 + 3*m^2 + 2*m as the product of the three unknown integers in terms of m.  Of the three integers: m,n& o, we know the smallest was the one we referred to as m, so the cube of the smallest is m^3.

 

Therefore you should be able to get an expression that equals 85 but is only a quadratic.  There are several methods for finding the solutions to a quadratic. I'd be glad to teach them to you if you don't know them yet.

 

Please respond if you get stuck on solving the rest of this question, or explain how far you get.

 

 

For the second problem, we can call the price of a dozen eggs E (or x if you prefer), and the price of a loaf of bread B (or y if you prefer).

The information says that

3E+5B = $6.90

4E+2B = $5.70.

 

Such systems of two equations using substitution, elimination, various methods using matrices, or graphing.  Which would you prefer?

 

For the last question, we know that both the amount of salt and the total volume are conserved.

Let's call the amount (in mL) of 20% solution T (for twenty) and the amount of Eighteen Percent solution (in mL) E (for eighteen).

 

So for the salt equation: 15%*100mL = 20%*T + 18%*E + 10% * 40mL

For the volume equation: 100 mL = T + E + 40 mL

 

We can multiply both sides of the first equation by 100 to get rid of all the percents (optional)

1500 = 20T+18E+400 or (subtracting 400 from both sides) 1100 = 20T + 18E

 

From the second equation, subtract 40 mL from both sides:

60 mL = T + E.

 

I'll start you off solving this system by substitution.

From the last equation, E = 60 - T

Therefore plug the right hand side (60-T) wherever we see E in 1100 = 20T+18E

1100 = 20T+18(60-T)

Finish solving this for T and let me know how it went. Be sure to check your answer in the salt equation and the volume equation.