algebra< odd number

Question

1. the ages is  of juan and his sister add up to half their father's age> if juan is 3 yrr than older than his sister and 27 yr younger than his father how many yrs old is juan

 

2.altogether, 6 chicks eat 8 cups of grains in 3 days. how many cups ofgrain will 3 chicks eat in 9 days

 

3.an odd  whole number between 600 and 800 is divisible by 7 and9> what is the sum of its digits.

 

 

4.what is the value of 1-2+3-4+5-6+...+53- 54+55

Answer 1

1. The ages of juan and his sister add up to half their father's age. If juan is 3 yrs older than his sister and 27 yrs younger than his father, how many yrs old is juan?

J + S = F/2
J = S + 3 => S = J - 3
J = F - 27 => F = J + 27

J + J - 3 = (J + 27)/2
2J - 3 = (J + 27)/2
4J - 6 = J + 27
3J = 33
J = 11

2. altogether, 6 chicks eat 8 cups of grains in 3 days. how many cups of grain will 3 chicks eat in 9 days.

rate = (8cups/3days)/6chicks = 8/18 = (4/9 cup/day)/chick

((4/9 cup/day)/chick) * (3 chicks) * (9 days) = 12 cups

3. an odd whole number between 600 and 800 is divisible by 7 and 9; what is the sum of its digits.

2n+1 is an odd whole number if n is a whole number.
To be divisible by 7 and 9 both 7 and 9 must be factors, 7*9*(2n+1) is such a number.
600 < 7*9*(2n+1) < 800
600/63 < (2n+1) < 800/63
200/21-21/21 < 2n < 800/63-63/63
179/21 < 2n < 737/63
179/42 < n < 737/126
4.26190476190476 < n < 5.84920634920635
n = 5
2n+1 = 11
63*11 = 693 is the number and sum of its digits is 18

4.what is the value of 1-2+3-4+5-6+...+53- 54+55 = S

         n =     1     2     3   …   26    27
S_even = -   2 -   4 -  6 - … - 52 - 54
S_even = - 54 - 52 - 50 - … -  4 -   2
2*S_even = -56(27)
S_even = -28(27)

       n =   1       2     3    …    27    28
S_odd =   1 +   3 +  5 + … + 53 + 55 = 2n-1
S_odd = 55 + 53 + 51 + … +  3 + 1
2*S_odd = 56(28)
S_odd = 28(28)

S = S_odd + S_even = 28(28) - 28(27) = 28(28-27) = 28

Answer 2

2) If a number is divisible by 9 and by 7 it is also divisible by 9*7=63.

10*63=630 but 630 is even

630+63=693 which is odd, divisible by 9, and also divisible by 7; so add the digits 6+9+3=18

693+63=756 which is even and 756+63=819 which is greater than 800

the answer is 693 and the sum again is 6+9+3=18

Answer 3

2. If 6 chicks eat 8 cups in three days, then each chick eats 8/(3x6) per day = .44 cups/day.

 

3 chicks will eat in 9 days .44 x 3 x 9 = 12 cups

 

3. I would brute force this, and generate an Excel list of all odd numbers between 600 and 800 that are wholly divisible by 63.  Shouldn't take that long.

 

4. Again, you could use Excel to do the problem.  But the easiest way is to see this not as individual numbers being added, but number pairs.  The first pair is 1+55.  The second pair is 2+54.  Each pair will add to 56.  You will have 27 pairs of numbers, each totaling 56, plus the unpaired number 28 left over.

 

So you have (27 x 56) + 28.

 

Oops.  What I wrote above holds if the numbers are all ADDED, not subtracted.  Arthur's comment below is correct.

Answer 4

 J - Juan's age

 

  F - Father's age

 

  S - Sister's age

 

  J + S = 1/2 F    (1)

 

  J = 3 + S         ( 2)

 

  J = F - 27       ( 3)

 

   Substitute J from ( 3) into ( 1) & ( 2)

 

    3 + S + S = 1/2 F

 

      3 + S =  F -27

  

       6 +4S = 30 +s

 

     3S = 24   

 

      S = 8      J = 11   F = 38