Guess and check math problems?

Question

1. Plato has 58 coins in nickels, dimes, and quarters, The number of nickels is three less than twice the number of dimes. The total value of the coins is $7.40. How many of each type of coin did he use?

 

2. There are nine boys to every ten girls in a particular high school. There are 2622 students at the school. How many girls are there?

 

3. Ronnie's age plus the square of Alan's age is 2240. Alan's age plus the square of Ronnie's age is 1008. how old are Ronnie and Adan?

Answer 1

If you post each question separately, you'll get more answers.  

 

FIRST PROBLEM - each sentence gives us one of these three equations

 

N + D + Q = 58

 

N = 2*D - 3

 

5*N + 10*D + 25*Q = 740

 

To solve, substitute the middle equation into the first one to eliminate the N's, then rearrange the first one to get Q in terms of D.  Finally, in the bottom equation, substitute the middle one for the N and the top one for the Q, adn solve for the value of D.  Now substitute back and check.  

 

As a "guess and check", you'd pick some arbitrary number of dimes, say 10, then use the middle to calculate the number of nickels (20-3=17) and the top to calculate the quarters (58-27=31).  Then you'd multiply out the value and see if you were over or under.  (over).  If you're over, then increase the number of dimes and try again, which will drop the number of quarters and reduce the value.     

 

 

SECOND PROBLEM

 

G = B * 10/9

 

G + B = 2622

 

Substitute the first into the second, combine the B's, then cross multiply to get the answer.

 

As a guess and check, you can start by thinking that the number of girls is just a little bigger than the number of boys, so you could start with a guess that's just over half the total population.  Half of 2622 is around 1300.  9/10 of that is 1170 for a total of 2470.  there's 152 left, so do the same thing again.  Half is 76, round up to 80, times 9/10 is 72 for a totalof 152. Check.  Girls is 1300+80 for 1380, boys is 1170+72 = 1242.  total is 1380+1242 = 2622.

 

 

THIRD PROBLEM

 

R + A^2 = 2240

 

A + R^2 = 1008

 

As an algebra problem, you'd solve for A in the second, substitute it in the first, clean it up and try to find the roots. 

 

As a guess and check, what do we know? Well, Ronnie's age is a lot less than Alan's, because the square root is a lot less.    

 

Also, neither one of them can be 50 years old, because then the square would be 2500. 

 

So, let's guess that Alan is 40.  How old must Ronnie be?  2240-1600=640.  Nope.  

Okay, 45, squared is 2025.  2240 - 2025 = 215.  Not likely.

Okay, 47, squared is 2209.  2240 - 2209 = 31.  Looks promising

31 squared is 961. 961 + 47 = 1008.  Check.