Lesson 2: More Challenging Ratio Problems

Property

If two quantities $A$ and $B$ are in the ratio $a:b$, we can express them as $A = ak$ and $B = bk$. When the quantities change by amounts $x$ and $y$ respectively, a new ratio $c:d$ is formed. This relationship can be modeled with the equation:

Examples

• The ratio of apples to bananas is $2:3$. After adding 5 apples and 5 bananas, the new ratio is $3:4$. The original number of fruits can be found by solving $\frac{2k+5}{3k+5} = \frac{3}{4}$, which gives $k=5$. Thus, there were originally $2(5)=10$ apples and $3(5)=15$ bananas.
• The number of boys and girls in a club are in the ratio $7:4$. When 11 more boys and 11 more girls join, the ratio becomes $4:3$. To find the original number of members, we set up the equation $\frac{7k+11}{4k+11} = \frac{4}{3}$. Solving this yields $k=11$, so there were originally $7(11)=77$ boys and $4(11)=44$ girls.

Explanation

This skill addresses ratio problems where the initial quantities undergo a change. By representing the original amounts with a common multiplier, $k$, we can set up an algebraic equation for the "after" scenario. Solving this equation for $k$ allows us to determine the original quantities. This method is often called a "before and after" problem and is a powerful way to handle dynamic ratio situations.

Property

If two variables $x$ and $y$ are related by the equation $ax = by$, where $a$ and $b$ are non-zero constants, their ratio is given by:

This means the ratio $x:y$ is equal to $b:a$.

Examples

• If $5a = 8b$, then the ratio of $a$ to $b$ is $\frac{a}{b} = \frac{8}{5}$, or $a:b = 8:5$.
• Given the equation $3x - 4y = 0$, we can rearrange it to $3x = 4y$. The ratio of $x$ to $y$ is $\frac{x}{y} = \frac{4}{3}$, or $x:y = 4:3$.
• If $\frac{1}{2}p = \frac{3}{4}q$, then the ratio of $p$ to $q$ is $\frac{p}{q} = \frac{3/4}{1/2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2}$.

Explanation

This skill involves rearranging a linear equation with two variables to find the ratio between them. To find the ratio $\frac{x}{y}$ from an equation like $ax = by$, you can isolate the fraction $\frac{x}{y}$ by dividing both sides of the equation. First divide by $y$, and then divide by $a$ to find the resulting numerical ratio. This technique is useful for working backwards from an equation to the ratio it represents, a common step in solving complex ratio problems.

Property

For chained ratios where $a:b = m:n$ and $b:c = p:q$, set up the system:

To find $a:b:c$, make the $b$ values equal by finding a common multiple.

Property

The ratio of the angular speed of a clock''s minute hand to its hour hand is $12:1$. This is because the minute hand completes a full $360^\circ$ rotation in 60 minutes (a speed of $6^\circ$ per minute), while the hour hand completes a $360^\circ$ rotation in 12 hours or 720 minutes (a speed of $0.5^\circ$ per minute).

Examples

• In the time it takes the hour hand to move $5^\circ$, the minute hand moves $12 \times 5^\circ = 60^\circ$.
• At 3:00, the hands are $90^\circ$ apart. To find when they will next coincide, we solve for the time $t$ in minutes when the minute hand has "caught up" the $90^\circ$ lead plus the distance the hour hand has moved. The relative speed of the minute hand to the hour hand is $6 - 0.5 = 5.5^\circ$ per minute. The time taken is $t = \frac{90^\circ}{5.5^\circ/\text{min}} = \frac{180}{11}$ minutes.

Explanation

This skill applies ratio concepts to the proportional movement of clock hands. The minute hand moves 12 times faster than the hour hand. Understanding this constant $12:1$ speed ratio is key to solving problems about the relative positions of the hands. By calculating the angular speeds of each hand, you can set up equations to determine when the hands will align, be opposite, or form a specific angle.