Proportional Reasoning-2Class 8 Mathematics Important Points

Two ratios $a:b$ and $c:d$ are proportional if their cross-products are equal, meaning $a \times d = b \times c$. Alternatively, the ratios of their corresponding terms are equal, $\frac{a}{b} = \frac{c}{d}$.

Two ratios $a:b$ and $c:d$ are proportional if their fractional forms are equal, $\frac{a}{b} = \frac{c}{d}$. This can be verified using the cross-multiplication method, where the relationship must satisfy $a \times d = b \times c$.

Two ratios with multiple terms, such as $a:b:c$ and $p:q:r$, are proportional if the ratios of their corresponding terms are equal. This is expressed as $\frac{a}{p} = \frac{b}{q} = \frac{c}{r} = k$, where $k$ is a constant factor.

If two multi-term ratios such as $a:b:c$ and $p:q:r$ are proportional, it means the ratio of their corresponding terms is constant. This is expressed as $\frac{a}{p} = \frac{b}{q} = \frac{c}{r}$.

To divide a quantity $X$ in the ratio $a:b:c$, first find the sum of the ratio parts $S = a+b+c$. The resulting shares are then calculated as $\frac{a}{S} \times X$, $\frac{b}{S} \times X$, and $\frac{c}{S} \times X$.

To divide a total quantity $Q$ in the ratio $a:b:c$, first find the sum of the ratio parts, $S = a+b+c$. The resulting parts are $\frac{a}{S} \times Q$, $\frac{b}{S} \times Q$, and $\frac{c}{S} \times Q$.

A map scale, or Representative Fraction (RF), like $1:N$ means that 1 unit of distance on the map represents $N$ units of actual distance on the ground. For example, a scale of $1:50,000$ means $1$ cm on the map equals $50,000$ cm on the ground.

A map scale, or Representative Fraction (RF), is a ratio like $1:N$. This means 1 unit of distance on the map represents $N$ units of actual distance on the ground. For example, with a scale of $1:50,000$, $1$ cm on the map equals $50,000$ cm (or $0.5$ km) in reality.

To create a pie chart, the total angle of a circle, $360^\circ$, is divided in the given ratio of the data. The angle for a specific item is calculated using the formula: $\text{Angle} = \frac{\text{Value of Item}}{\text{Total Value}} \times 360^\circ$.

In a pie chart, the angle of each sector is proportional to the value it represents. The formula to find the angle is: Central Angle $= \frac{\text{Value of the component}}{\text{Total value}} \times 360^\circ$. The sum of all angles must be $360^\circ$.

Two quantities $x$ and $y$ are in direct proportion if they change by the same factor, meaning their ratio remains constant. For any two pairs of values $(x_1, y_1)$ and $(x_2, y_2)$, we have $\frac{x_1}{y_1} = \frac{x_2}{y_2} = k$, where $k$ is a constant.

Two quantities $x$ and $y$ are in direct proportion if they change by the same factor. As one increases, the other increases. Their ratio is constant: $\frac{x}{y} = k$, which implies $\frac{x_1}{y_1} = \frac{x_2}{y_2}$.

Two quantities $x$ and $y$ are in inverse proportion if an increase in one causes a proportional decrease in the other. Their product remains constant, so $x \times y = k$ for any pair of values.

Direct proportion is seen in scenarios like: the more items you buy, the higher the total cost (at a fixed price per item); the farther you travel at a constant speed, the more time it takes.

If $(x_1, y_1)$ and $(x_2, y_2)$ are pairs of values in an inverse proportion, then their products are equal: $x_1 y_1 = x_2 y_2$. This can be rearranged to find an unknown value, for example, $y_2 = \frac{x_1 y_1}{x_2}$.

Two quantities $x$ and $y$ are in inverse proportion if one increases while the other decreases by the same factor. Their product is constant: $x \times y = k$, which implies $x_1 y_1 = x_2 y_2$.

Common examples of inverse proportion include the number of workers and the time to finish a job, or the speed of a vehicle and the time taken to cover a fixed distance. As one quantity increases, the other decreases.

Inverse proportion occurs in situations like: increasing the number of workers to decrease the time to finish a job; increasing speed to decrease the time taken to cover a fixed distance.

If $(x_1, y_1)$ and $(x_2, y_2)$ are pairs of values in an inverse proportion, you can find an unknown value using the equation $x_1 y_1 = x_2 y_2$. For example, to find $y_2$, the formula is $y_2 = \frac{x_1 y_1}{x_2}$.

To find the time taken for two individuals to complete a task together, first find their individual work rates. If person A takes $t_A$ hours and person B takes $t_B$ hours, their combined work rate is $(\frac{1}{t_A} + \frac{1}{t_B})$ of the job per hour.

The sum of angles in any triangle is $180^\circ$. To find the angles of a triangle with a given ratio $a:b:c$, you divide $180^\circ$ into parts corresponding to that ratio.

To determine the type of proportion, ask yourself: 'If I increase quantity A, does quantity B increase or decrease?' If B increases, it is direct proportion. If B decreases, it is inverse proportion.

A triangle can only be constructed if the sum of the lengths of any two sides is greater than the length of the third side. This rule must also apply to a ratio representing side lengths. For example, a ratio of $1:3:5$ cannot form a triangle because $1+3$ is not greater than $5$.

The number of workers and the time taken to complete a fixed amount of work are generally in inverse proportion. For example, if you double the number of workers, the time required to complete the job is halved.

Work rate is the amount of work done per unit of time. If a person can complete a job in $n$ days, their work rate is $\frac{1}{n}$ of the job per day. This concept is key to solving problems where multiple people work together.

If person A takes $x$ hours and person B takes $y$ hours to do a job, their individual work rates are $\frac{1}{x}$ and $\frac{1}{y}$ per hour. Their combined rate is $\frac{1}{x} + \frac{1}{y}$. The time to finish the job together is the reciprocal of this combined rate.