XI.

4. If $4$ yards of velvet and $3$ yards of silk are sold for $$33$, and $5$ yards of velvet and $6$ yards of silk for $$48$, what is the price per yard of the velvet and of the silk?

5. If $7$ bushels of wheat and $10$ of rye are sold for $$15$, and $4$ bushels of wheat and $5$ of rye are sold for $$8$, what is the price per bushel of the wheat and of the rye?

6. If $12$ pounds of tea and $4$ pounds of coffee cost $$7$, and $4$ pounds of tea and $12$ pounds of coffee cost $$5$, what is the price per pound of tea and of coffee?

7. Six horses and $7$ cows can be bought for $$1000$, and $11$ horses and $13$ cows for $$1844$. Find the value of a horse and of a cow.

Exercise 67.

Ex. A certain fraction becomes equal to $\frac{1}{2}$ if $2$ is added to its numerator, and equal to $\frac{1}{3}$ if $3$ is added to its denominator. Find the fraction.

lint Let xy = the required fraction. lint Then x + 2y = 12, lintand xy + 3 = 13.

The solution of these equations gives $7$ for $x$, and $18$ for $y$.

Therefore the required fraction is $\frac{7}{18}$.

1. If the numerator of a certain fraction is increased by $2$ and its denominator diminished by $2$, its value will be $1$. If the numerator is increased by the denominator and the denominator is diminished by $5$, its value will be $5$. Find the fraction.

2. If $1$ is added to the denominator of a fraction, its value will be $\frac{1}{2}$. If $2$ is added to its numerator, its value will be $\frac{3}{5}$. Find the fraction.

3. If $1$ is added to the numerator of a fraction, its value will be $\frac{1}{5}$. If $1$ is added to its denominator, its value will be $\frac{1}{7}$. Find the fraction.

4. If the numerator of a fraction is doubled and its denominator diminished by $1$, its value will be $\frac{1}{2}$. If its denominator is doubled and its numerator increased by $1$, its value will be $\frac{1}{7}$. Find the fraction.

5. In a certain proper fraction the difference between the numerator and the denominator is $15$. If the numerator is multiplied by $4$ and the denominator increased by $6$, its value will be $1$. Find the fraction.

Exercise 68.

The expression $64$ means $60+4$, that is, $10$ times $6+4$, and has for its digits $6$ and $4$. If the digits were unknown and represented by $x$ and $y$, the number would be represented by $10x+y$.

Ex. The sum of the two digits of a number is $10$, and if $18$ is added to the number, the digits will be reversed. Find the number.