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# Help

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1.Jerry writes down all the odd numbers 1, 3, 5, 7,  up to 999. How many numbers does he write down?

2.There are 100 students in South High School. South High School offers only Chinese and Spanish. There are 8 more students in Chinese than in Spanish, and every student takes at least one language. If 37 students take only Spanish, then how many take both languages?

3.If you write out the numbers from 1 through 60 (including 1 and 60), how many numbers have a "5" in them?

4.There are 40 animals at a petting zoo, all of which are goats or rabbits, and each animal is either an adult or a child. There are twice as many adult rabbits as adult goats, and three times as many child goats as adult goats. If there are an equal amount of adult rabbits as child rabbits, how many adult goats are there at the petting zoo?

5.In the Gregorian calendar, every year which is divisible by  is a leap year, except for years which are divisible by 100; those years are only leap years if they're divisible by 400.(This may seem complicated, but the calendar is carefully designed to keep the average number of days per year very close to the number of days in one complete orbit of the Earth.)Assuming we keep using the Gregorian calendar, how many leap years will there be between 2001 and 2999?

Apr 8, 2019
edited by Saketh  Apr 8, 2019

#1
+3

1.Jerry writes down all the odd numbers 1, 3, 5, 7,  up to 999. How many numbers does he write down?

Let's look at something more simple....

What if he wrote down   1, 3, 5, 7 , 9   ???

Note that the number of odd integers he writes is    [ last odd + 1 ] / 2  =  [  9 + 1 ] / 2   =  10 / 2 = 5

So...this implies that the number of odds written is   [ 999 + 1 ] / 2  = 1000 / 2  =   500   Apr 8, 2019
#2
+3

3.If you write out the numbers from 1 through 60 (including 1 and 60), how many numbers have a "5" in them?

5 have a  "5" as a units digit  [ not including 55]

10 more  [ 50 - 59 ]  contain a "5"

So

15  contain a "5"   Apr 8, 2019
#3
+2

4.There are 40 animals at a petting zoo, all of which are goats or rabbits, and each animal is either an adult or a child. There are twice as many adult rabbits as adult goats, and three times as many child goats as adult goats. If there are an equal amount of adult rabbits as child rabbits, how many adult goats are there at the petting zoo?

CR = Child rabbits

CG = Child goats

2AG = AR

3AG = CG

AR = CR

AR + AG + CR + CG  = 40        substitute

2AG  + AG + AR+ CG  = 40

2AG + AG + 2AG + 3AG  = 40

8AG  =  40

AG = 5   Apr 8, 2019
#4
+2

5.In the Gregorian calendar, every year which is divisible by  is a leap year, except for years which are divisible by 100; those years are only leap years if they're divisible by 400.(This may seem complicated, but the calendar is carefully designed to keep the average number of days per year very close to the number of days in one complete orbit of the Earth.)Assuming we keep using the Gregorian calendar, how many leap years will there be between 2001 and 2999?

We have   998 years

floor [ 998/4 ]  = 249  possible leap years

But  the years 2200, 2300, 2500, 2600, 2700 and 2900 do not qualify because they are not evenly divisible by 400

So

249 - 6  =

243 leap years   Apr 8, 2019
#5
+2

2.There are 100 students in South High School. South High School offers only Chinese and Spanish. There are 8 more students in Chinese than in Spanish, and every student takes at least one language. If 37 students take only Spanish, then how many take both languages?

These always give me some trouble....!!!!

Let N be the number that take both

37 + N   =  number who take Spanish

37 + N + 8  = 45 + N  = number who take Chinese

So

Number ( Spanish + Chinese)  = Number(Spanish) + Number (Chinese)  - Number (Both)    = 100

(37 + N)  +  (45 + N )  - N   =  100

37 + 45 + N = 100

82 + N  = 100

N = 18  =  number that take both  !!!!

So

37 take only Spahish , 18 take both and 45 take only Chinese  = 100   Apr 8, 2019