On monday, Tanya counted the number of 10-cent coins, 20-cent coins and 50-cent coins in her piggy bank. The number of 10-cent coins was 24 more than the number of 50-cent coins. On tuesday, Tanya spent 3 over
5 of the 10-cent coins, 1 over 4 of the 20-cent coins and 18 50-cent coins. Tanya then found that she had an equal number of 10-cent coins and 20-cent coins. The number of 50-cent coin now formed 40% of all the coins she had left.
(a)Tanya had how many 10-cent coins at first?
(b)How much money did Tanya spent in all?
Addendum......I believe I spotted Melody's error
She has
0.72X - 16.8 =X - 42
16.8 + 42 = x - 0.72X
But the 16.8 should be negative
So....it should be....
-16.8 + 42 = 0.28X
25.2 = 0.28X
X = 90 = the number of 10 cent coins, originally.....
And our answers would now agree....
Hey.....no biggie......!!!!
On monday, Tanya counted the number of 10-cent coins, 20-cent coins and 50-cent coins in her piggy bank. The number of 10-cent coins was 24 more than the number of 50-cent coins. On tuesday, Tanya spent 3 over
5 of the 10-cent coins, 1 over 4 of the 20-cent coins and 18 50-cent coins. Tanya then found that she had an equal number of 10-cent coins and 20-cent coins. The number of 50-cent coin now formed 40% of all the coins she had left.
(a)Tanya had how many 10-cent coins at first?
This one is really confusing is it not :)
Let the number of tens be X, the number of twenties be Y and the number of fifties be Z
Z+24=X
X-24=Z
-----
now after her spending spree this is what what she has left
(2/5)X+(3/4)y+(Z-18) coins
and
(2/5)X=(3/4)Y
so Y = 0.4X/0.75 = (8/15)X and from before X-24=Z
also
0.4((2/5)X+(3/4)y+(Z-18) )=Z-18
The first thing we want is X so I will get all this in terms of X and solve and hopefully the answer is a whole number. :)
0.4((2*(2/5)X+(X-24-18))=X-24-18
0.4((0.8X+(X-42))=X-42
0.4(1.8X-42)=X-42
0.4*1.8X - 0.4*42=X-42
0.72X - 16.8 =X - 42
-16.8+42 =X - 0.72X
25.2=0.28X
X=90
so there was originally 90 ten cent peices
(b)How much money did Tanya spent in all?
X=90
Y=(8/15)X = 8/15*90 = 48
She spent (3/5)*X = (3/5)*90 = 54 ten cent coins = $5.40
She spent (1/4)*Y = 0.25*48 = 12 twenty cent coins = $2.40
She spent 18 fifty cent coins = $9
Total spent = 5.40+2.40+9= $16.80
I fixed this after CPhill told me, by Private message, of a careless error. Thanks Chris
I only just realized that he has posted a proper answer underneath then whited it out.
You don't need to do that Chris - I shall unwhite it. It is a good answer :)
I get a different answer from Melody on this one.....
Let x be the number of 50 cent coins in the beginning
Let y be the number of 20 cent coins in the beginning
Let x + 24 be the number of 10 cent coins in the beginning
And after the spree we have
x - 18 = the number of 50 cent coins
(2/5)(x + 24) = the number of 10 cent coins = the number of 20 cent coins
So the total coins left after the spree = 2(2/5)(x + 24) + ( x -18) = (4/5)(x + 24) + (x - 18)
And 40% of these = the number of 50 cent coins left .... so we have
.40[ (4/5)(x + 24) + (x - 18) ] = (x - 18)
(2/5)[ (4/5 (x + 24) + (x - 18) ] = (x - 18)
[ (4/5 (x + 24) + (x - 18) ] = (5/2)(x - 18) multiply through by 10
8(x + 24) + 10( x - 18) = 25(x - 18)
8x + 192 + 10x - 180 = 25 x - 450 simplify
18x + 12 = 25x - 450
462 = 7x divide by 7 on each side
x = 66 this is the number of 50 cent coins in the beginning
And x + 24 = (66 + 24) = 90 were 10 cent coins in the beginning
And notice that (3/4)y = (2/5) (x + 24)... so y = (8/15)(66 + 24) = (8/15)(90) = 48 coins
So....there were 48 20 cent coins originally
Now.....let's check the math....
The total number of coins after the spree =
[(4/5)(66 + 24) + (66 - 18) ] = 120
And (66 - 18) = 48 of these were 50 cent coins
So..... 48 / 120 = 40% were 50 cent coins
And after the spree....there were an equal number of 10 and 20 cent coins.....check ...(2/5)(90) = (3/4)(48) = 36 each
So the total spent = (3/5)(90)(.10) + (1/4)(48)(.20) + (18)(.50) = $16.80
I believe Melody has a slight error
She gets 210 10 cent coins at the start .....and 2/5 of these are left = 84 left
And this is equal to the number of 20 cent coins left
And she calculates the number of 50 cent coins left = 210 - 42 = 168
So....the total coins left = [ 84 + 84 + 168] = 336 coins left
But 168 / 336 = 50% of the coins left are 50 cent coins......and this is too much
Can anyone reconcile these differences ???
Addendum......I believe I spotted Melody's error
She has
0.72X - 16.8 =X - 42
16.8 + 42 = x - 0.72X
But the 16.8 should be negative
So....it should be....
-16.8 + 42 = 0.28X
25.2 = 0.28X
X = 90 = the number of 10 cent coins, originally.....
And our answers would now agree....
Hey.....no biggie......!!!!