XI.

lint Let x = the tens' digit, lintand y = the units' digit. lint Then 10x + y = the number. [1] lint Hence x + y = 10, (1) lintand 10x + y + 18 = 10y + x. (2) [1] lint From (2), 9x - 9y = -18, lintor x - y = -2. (3) [1] lint Add (1) and (3), 2x = 8, lintand therefore x = 4. lint Subtract (3) from (1), 2y = 12, lintand therefore y = 6.

Therefore the number is $46$.

1. The sum of the two digits of a number is $9$, and if $9$ is added to the number, the digits will be reversed. Find the number.

2. A certain number of two digits is equal to eight times the sum of its digits, and if $45$ is subtracted from the number, the digits will be reversed. Find the number.

3. The sum of a certain number of two digits and the number formed by reversing the digits is $132$, and the difference of these numbers is $18$. Find the numbers.

4. The sum of the two digits of a number is $9$, and if the number is divided by the sum of its digits, the quotient is $6$. Find the number.

Exercise 69.

Ex. A sum of money, at simple interest, amounted to $$2480$ in $4$ years, and to $$2600$ in $5$ years. Find the sum and the rate of interest.

lint Let x = the number of dollars in the principal, lintand y = the rate of interest.

The interest for one year is ${\displaystyle \frac{y}{100}}$ of the principal; that is, ${\displaystyle \frac{xy}{100}}$. For $4$ years the interest is ${\displaystyle \frac{4xy}{100}}$, and for $5$ years ${\displaystyle \frac{5xy}{100}}$. The amount is principal $+$ interest,

lintor x + 4xy100 = 2480. x + 5xy100 = 2600. lint Hence 100x + 4xy = 248,000. (1) 100x + 5xy = 260,000. (2) [1] Divide (1) by $4$ and (2) by $5$, and we have 25x + xy = 62,000 20x + xy = 52,000 [1] lint Subtract, 5x + xy = 10,000. [1] lint Therefore 5xx + xy = 10,0002000.

Substitute the value of $x$ in (1), $y=6$.

Therefore the sum is $$2000$, and the rate $6$

1. A sum of money, at simple interest, amounted in $5$ years to $$3000$, and in $6$ years to $$3100$. Find the sum and the rate of interest.

2. A sum of money, at simple interest, amounted in $10$ months to $$1680$, and in $18$ months to $$1744$. Find the sum and the rate of interest.

3. A man has $$10,000$ invested, a part at $4$%, and the remainder at $5$%. The annual income from his $4$% investment is $$40$ more than from his $5$% investment. Find the sum invested at $4$% and at $5$%.

Exercise 70.