**Keane bought some marbles and gave half of them to Leon. Leon then bought some stamps and gave half of the to Keane. Keane used 5 stamps and Leon gave away 11 marbles. The ratio of the number of stamps to the number of marbles Keane had left then became 1:7 and the ratio of the number of stamps to the number of marbles Leon had left became 1:5. How many stamps did Leon buy?
Solution**

M ----- Marbles

S ----- Stamps

Step 1

Keane bought marbles and gave half to Leon,

Keane ----- M

Leon ----- M

Both have same number of marbles.

Step 2

Leon bought stamps and gave half to Keane,

Keane ----- M , S

Leon ----- M , S

Now both have same number of marbles and stamps

Step 3

Keane used 5 stamps, Leon gave away 11 marbles,

Keane ----- M , [S - 5]

Leon ----- [M - 11] , S

Now Keane has 5 stamps less, and Leon 11 marbles less

Step 4

Keane's no. of stamps to no. of marbles now is 1:7,

M ----- (7 units)

[S - 5] ----- (1 unit)

Now Keane has 7 times as many marbles as stamps.

Step 5

Leon's no. of stamps to no. of marbles now is 1:5,

[M - 11] ----- (5 parts)

S ----- (1 part)

Now Leon has 5 times as many stamps as marbles.

Step 6

1 unit of Keane's stamps + 5 ----- 1 part of Leon's stamps

1 unit + 5 ----- 1 part

Keane gave away 5 stamps, so to have an equal no. of stamps as Leon now, we add back the 5 stamps Keane gave away.

Step 7

7 units of Keane's marbles ----- 5 parts of Leon's marbles + 11

7 units ----- 5 parts + 11

Leon gave away 11 marbles, so to have an equal no. of marbles as Keane, we add back the 11 marbles Leon gave away.

Step 8

Comparing marbles with stamps

(Stamps) 1 unit + 5 ----- 1 part (refer to Step 6)

(Marbles) 7 units ----- 5 parts + 11 (refer to Step 7)

Step 9

Multiply stamps by 5

(Stamps) 5 units + 25 ----- 5 parts

(Marbles) 7 units --> 5 parts + 11

(Stamps) 5 units ----- 5 parts - 25

(Marbles) 7 units ----- 5 parts + 11

Step 10

(marbles) - (stamps)

7 units - 5 units --> 5 parts + 11 - 5 parts - (-25)

2 units --> 11 + 25

2 units --> 36

1 unit --> 18

Step 11

1 unit ----- S - 5 (refer to Step 4)

18 ----- S - 5

S ----- 18 + 5 = 23

This is the no. of stamps Leon had after giving half away.

Step 12

Leon bought,

2 x 23 = 46

**Answer: 46 stamps**