Math Help - more problems..

1. more problems..

find positive integers x,y,z such that 28x+30y+31z=365

use exact calculation (no calculators or decimals allowed) to decide which is larger:
(a) 28% of 75 or 75% of 28? (b) 76% of 27 or 57% of 36?

the 1st July is a beautiful date, since the shortened form of the year number is equal to the product of the day number and the month number. 1.7.70. how many years in the 21st century have no beautiful dates at all?

there are more but some have diagrams so i cannot type them out,,

thanks u soo much!! xxxx

2. Originally Posted by mich13
find positive integers x,y,z such that 28x+30y+31z=365
Solve for one of the variables, say z.
$z = \frac{365 - 28x - 30y}{31}$

Now we need to find an x, y pair such that the numerator is divisible by 31. There is an advanced method to help you with this, but the simplest thing to do is guess and check:
$(x, y) = (1, 1) \implies z = \frac{301}{31}$

$(x, y) = (1, 2) \implies z = \frac{277}{31}$

etc.

I get
$(x, y) = (1, 4) \implies z = 7$

A less trivial version:
$(x, y) = (2, 1) \implies z = 9$

I believe this covers the entire solution set.

-Dan

3. Originally Posted by mich13
use exact calculation (no calculators or decimals allowed) to decide which is larger:
(a) 28% of 75 or 75% of 28? (b) 76% of 27 or 57% of 36?
a)
$28 \% = \frac{28}{100} = \frac{7}{25}$

$75 \% = \frac{75}{100} = \frac{3}{4}$

So 28% of 75 is:
$\frac{7}{25} \cdot 75 = 7 \cdot 3 = 21$
and
75% of 28 is
$\frac{3}{4} \cdot 28 = 3 \cdot 7 = 21$

So they are equal.

The others are done the same way.

-Dan

4. Originally Posted by mich13
the 1st July is a beautiful date, since the shortened form of the year number is equal to the product of the day number and the month number. 1.7.70. how many years in the 21st century have no beautiful dates at all?
I don't understand. 1.7.70. The year is 70, the day is 1 and the month is 7. But $1 \cdot 7 \neq 70$??

(Now October 7th, 1970 ==> 7.10.70 works.)

-Dan

5. im still a little confused but thank you SO SO SO SO SOOO much for taking your time to help me you must be some mathematical genius to get it that quickly!!
thanks xx
ps. how old r u???????

6. sorry i meant 1.7.07 typing error

7. Originally Posted by mich13
im still a little confused but thank you SO SO SO SO SOOO much for taking your time to help me you must be some mathematical genius to get it that quickly!!
thanks xx
ps. how old r u???????
Not a genius. Just been doing this for a while and I have a nice calculator to help me through the rough spots.

For the record, I'll be 37 in August. (sobs)

-Dan

8. $28 \% = \frac{28}{100} = \frac{7}{25}$

$75 \% = \frac{75}{100} = \frac{3}{4}$

So 28% of 75 is:
$\frac{7}{25} \cdot 75 = 7 \cdot 3 = 21$
and
75% of 28 is
$\frac{3}{4} \cdot 28 = 3 \cdot 7 = 21$

So they are equal.

The others are done the same way.

how did you get:
So 28% of 75 is:
$\frac{7}{25} \cdot 75 = 7 \cdot 3 = 21$
and
75% of 28 is
$\frac{3}{4} \cdot 28 = 3 \cdot 7 = 21$
???????????????????????
-mich xxx

9. Originally Posted by mich13
find positive integers x,y,z such that 28x+30y+31z=365

Do you remember how many months have 31, 30 and 28 days in a year?

10. Originally Posted by mich13
$28 \% = \frac{28}{100} = \frac{7}{25}$

$75 \% = \frac{75}{100} = \frac{3}{4}$

So 28% of 75 is:
$\frac{7}{25} \cdot 75 = 7 \cdot 3 = 21$
and
75% of 28 is
$\frac{3}{4} \cdot 28 = 3 \cdot 7 = 21$

So they are equal.

The others are done the same way.

how did you get:
So 28% of 75 is:
$\frac{7}{25} \cdot 75 = 7 \cdot 3 = 21$
and
75% of 28 is
$\frac{3}{4} \cdot 28 = 3 \cdot 7 = 21$
???????????????????????
-mich xxx
28% is the same as $\frac{28}{100}$ which is the same as:
$\frac{28}{100} = \frac{4 \cdot 7}{4 \cdot 25} = \frac{7}{25}$

Thus to find 28% of a number we multiply it by $\frac{7}{25}$. 28% of 75 becomes:
$\frac{7}{25} \cdot 75 = \frac{7 \cdot 75}{25} = \frac{7 \cdot 3 \cdot 25}{25} = 7 \cdot 3 = 21$.

-Dan

11. Originally Posted by mich13
use exact calculation (no calculators or decimals) to decide which is larger:
76% of 27 or 57% of 36?

pleeeeasssssse!
thanks xxx
76% is $\frac{76}{100} = \frac{19}{25}$

So 76% of 27 is:
$\frac{19}{25} \cdot 27 = \frac{19 \cdot 27}{25}$

57% is $\frac{57}{100}$

So 57% of 36 is:
$\frac{57}{100} \cdot 36 = \frac{57 \cdot 36}{100} = \frac{57 \cdot 9 \cdot 4}{25 \cdot 4} = \frac{57 \cdot 9}{25}$

So which is bigger?
$\frac{19 \cdot 27}{25}$ or $
\frac{57 \cdot 9}{25}$
?

They are both over the same denominator, so the question becomes which numerator is bigger?
$19 \cdot 27 = 19 \cdot (3 \cdot 9)$
or
$57 \cdot 9 = (3 \cdot 19) \cdot 9$?

Again we see that the two are equal since the factors are the same.

-Dan

12. ..

i get them all now, thanks! but i still dont understand the 'beautiful date' question... or how to do it!