The present age of a husband and wife is in the ratio $5:4$. After $6\text{ years}$ their ages will be in the ratio $6:5$. At the time of their marriage the ratio of their ages was $4:3$. How many years ago were they married?

Let the present age of the husband be $5x$ and the wife be $4x$.After $6\text{ years}$, their ages will be $(5x+6)$ and $(4x+6)$ respectively.According to the problem:
$$\frac{5x+6}{4x+6}=\frac{6}{5}$$
Cross-multiplying to solve for $x$:
$$5(5x+6)=6(4x+6)$$
$$25x+30=24x+36$$
$$25x-24x=36-30$$
$$x=6$$
Therefore, their present ages are:$\text{Husband’s present age}=5\times 6=30\text{ years}$$\text{Wife’s present age}=4\times 6=24\text{ years}$ Let them be married $y$ years ago. At the time of marriage, their ages were $(30-y)$ and $(24-y)$.The ratio of their ages at that time was $4:3$:
$$\frac{30-y}{24-y}=\frac{4}{3}$$
Cross-multiplying to solve for $y$:
$$3(30-y)=4(24-y)$$
$$90-3y=96-4y$$
$$4y-3y=96-90$$
$$y=6$$
Hence, they were married $6\text{ years ago}$.