Ex. 5. The middle digit of a number between 100 and 1000 is zero, and the sum of the other digits is 11. If the digits be reversed, the number so formed exceeds the original number by 495. Find it.
Let x represent the digit in the units' place ;
y represent the digit in the hundreds' place ;
then, since the digit in the tens' place is 0, the number will be represented by 100 y + x. [Art. 84, Ex. 4.]
And if the digits are reversed, the number so formed will be represented by 100 x + y.
100 x + y - (100 y + x) = 495, or 100x + y- 100y-x = 495; 99x- 99y = 495, that is, x - y = 5 (1).
Again, since the sum of the digits is 11, and the middle one is 0, we have x + y = 11 (2).
From (1) and (2) we find x = 8, y = 3.
Hence the number is 308.
EXAMPLES XVIII.
1. Find two numbers whose sum is 34, and whose difference is 10.
2. The sum of two numbers is 73, and their difference is 37: find the numbers.
3. One-third of the sum of two numbers is 14, and one-half of their difference is 4: find the numbers.
4. One-nineteenth of the sum of two numbers is 4, and their difference is 30: find the numbers.
5. Half the sum of two numbers is 20, and three times their difference is 18: find the numbers.
6. Six pounds of tea and eleven pounds of sugar cost $5.65, and eleven pounds of tea and six pounds of sugar cost $9.65. Find the cost of tea and sugar per pound.
7. Six horses and seven cows can be bought for $250, and thirteen cows and eleven horses can be bought for $461. What is the value of each animal ?
8. A, B, C, D have $ 290 between them ; A has twice as much as C, and B has three times as much as D; also C and D together have $ 50 less than A. Find how much each has.
9. A, B, C, D have $270 between them; A has three times as much as C, and B five times as much as D; also A and B together have $50 less than eight times what C has. Find how much each has.
