11. (10 points) Grandma Wang took a bag of coins to the bank to exchange for paper money. The bag contains one-fen, two-fen, five-fen, and ten-fen coins. The number of two-fen coins is $\frac{3}{5}$ of the number of one-fen coins, the number of five-fen coins is $\frac{3}{5}$ of the number of two-fen coins, and the number of ten-fen coins is $\frac{3}{5}$ of the number of five-fen coins minus 7. Grandma Wang exchanged for a whole number of yuan that is greater than 50 and less than 100. How many of each type of coin does she have?
【Analysis】Assuming that adding 7 more coins of 10 cents is 70 cents, at this point, the number of 10-cent coins is $\frac{3}{5}$ of the 5-cent coins. If we consider the number of 1-cent coins as the unit "1", the number of 10-cent coins is $\frac{3}{5} \times \frac{3}{5} \times \frac{3}{5}=\frac{27}{125}$ of the 1-cent coins, so the number of 1-cent coins must be a multiple of 125. Therefore, the minimum number of 2-cent coins is: $125 \times \frac{3}{5}=75$ coins, the number of 5-cent coins is: $75 \times \frac{3}{5}=45$ coins, and the number of 10-cent coins is: $45 \times \frac{3}{5}=27$ coins. The total amount of money is: $1 \times 125+2 \times 75+5 \times 45+10 \times 27=770$ (cents) $<5000$ cents, which does not meet the requirement; Since we added 7 coins, which is 70 cents, the total amount of paper money that Aunt Wang's brother received should be between 5070 and 10070; the last two digits must also be "70". Therefore, the total amount of money is: $770 \times 11=8470$ (cents), so the number of 1-cent coins is: $125 \times 11=1375$ (coins), the number of 2-cent coins is: $75 \times 11=825$ (coins), the number of 5-cent coins is: $45 \times 11=495$ (coins), and the number of 10-cent coins is: $27 \times 11-7=290$ (coins); according to this, we can solve the problem.

【Solution】Solution: Assuming that adding 7 more coins of 10 cents is 70 cents, at this point, the number of 10-cent coins is $\frac{3}{5}$ of the 5-cent coins. The number of 10-cent coins is $\frac{3}{5} \times \frac{3}{5} \times \frac{3}{5}=\frac{27}{125}$ of the 1-cent coins, so the number of 1-cent coins must be a multiple of 125. Therefore, the minimum number of 2-cent coins is: $125 \times \frac{3}{5}=75$ coins, the number of 5-cent coins is: $75 \times \frac{3}{5}=45$ coins, and the number of 10-cent coins is: $45 \times \frac{3}{5}=27$ coins. The total amount of money is: $1 \times 125+2 \times 75+5 \times 45+10 \times 27=770$ (cents) $<5000$ cents, which does not meet the requirement; Since we added 7 coins, which is 70 cents, the total amount of paper money that Aunt Wang's brother received should be between 5070 and 10070; the last two digits must also be "70".

Therefore, the total amount of money is: $770 \times 11=8470$ (cents), so the number of 1-cent coins is: $125 \times 11=1375$ (coins),

the number of 2-cent coins is: $75 \times 11=825$ (coins),
the number of 5-cent coins is: $45 \times 11=495$ (coins),
the number of 10-cent coins is: $27 \times 11-7=290$ (coins);
Answer: The number of 1-cent coins is 1375 coins, the number of 2-cent coins is 825 coins, the number of 5-cent coins is 495 coins, and the number of 10-cent coins is 290 coins.