Proportional Reasoning-1Class 8 Mathematics Important Points

A ratio $a:b$ compares two quantities, meaning for every 'a' units of the first quantity, there are 'b' units of the second. The numbers 'a' and 'b' are called the terms of the ratio.

A ratio $a:b$ compares two quantities, meaning for every 'a' units of the first quantity, there are 'b' units of the second. The numbers $a$ and $b$ are the terms of the ratio.

Two ratios are proportional if their terms change by the same multiplicative factor. For example, $60:40$ and $30:20$ are proportional because both terms of the first ratio are multiplied by $\frac{1}{2}$ to get the second.

Two ratios, $a:b$ and $c:d$, are proportional if they represent the same relationship. This is written as $a:b::c:d$ and means the ratios are equivalent.

To check if two ratios are proportional, reduce them to their simplest form by dividing both terms by their Highest Common Factor (HCF). If the simplest forms are identical, the ratios are in proportion.

To simplify a ratio, divide both terms by their Highest Common Factor (HCF). Ratios are proportional if their simplest forms are identical. For example, $90:60$ simplifies to $3:2$ by dividing by HCF 30.

The symbol '::' indicates that two ratios are proportional. The statement $a:b :: c:d$ reads as 'a is to b as c is to d' and means the ratios $a:b$ and $c:d$ are proportional.

In a proportion $a:b::c:d$, the terms change by the same multiplicative factor 'f'. Find the factor by dividing corresponding terms, for example f = rac{c}{a}, and then apply it to the other term, $d = b imes f$.

For two ratios to be in proportion as $a:b :: c:d$, the product of the extremes ($a$ and $d$) must equal the product of the means ($b$ and $c$). This gives the fundamental property $a \times d = b \times c$.

For a proportion $a:b::c:d$, the product of the outer terms (extremes) equals the product of the inner terms (means). This gives the formula $a imes d = b imes c$.

To find an unknown fourth term, $x$, in a proportion like $a:b :: c:x$, use the cross-multiplication rule to get the formula $x = \frac{b \times c}{a}$. This is also known as the 'Rule of Three'.

Using cross-multiplication, you can solve for any unknown term in $a:b::c:d$. For instance, to find the fourth term 'd', the formula is d = rac{b imes c}{a}.

To divide a total quantity $X$ into two parts in the ratio $m:n$, the first part is $\frac{m}{m+n} \times X$ and the second part is $\frac{n}{m+n} \times X$. The sum of the ratio terms ($m+n$) represents the total number of 'shares'.

To divide a total quantity $X$ in the ratio $m:n$, first find the sum of the ratio parts, $m+n$. The two parts are then rac{m}{m+n} imes X and rac{n}{m+n} imes X.

When setting up a proportion, it is critical to use consistent units for corresponding terms. For example, if one ratio involves minutes and kilometers, the second ratio must also use minutes and kilometers in the same order.

To divide ₹4500 in the ratio $2:3$, the sum of parts is $2+3=5$. The shares are rac{2}{5} imes 4500 = ext{₹}1800 and rac{3}{5} imes 4500 = ext{₹}2700.

Adding or subtracting the same number from both terms of a ratio will change the ratio and does not maintain proportionality. For example, $3:30$ is not proportional to $(3+9):(30+9)$, which is $12:39$.

When setting up a proportion, ensure that the quantities in each ratio are expressed in the same units. For example, compare minutes to minutes and grams to grams, not minutes to hours.

A ratio remains proportional only if its terms are multiplied or divided by the same non-zero number. Adding or subtracting the same number from both terms changes the ratio and breaks the proportion.

To compare two ratios, such as $a:b$ and $c:d$, convert them to fractions $\frac{a}{b}$ and $\frac{c}{d}$ and find a common denominator, or convert them to decimals. The larger value represents the 'stronger' or greater ratio.

In a proportion $a:b :: c:d$, you can find the factor of change by dividing the corresponding terms, such as $f = \frac{c}{a}$. The other term must change by the same factor, so $d = b \times f$.

The methods in this chapter apply to direct proportion, where if one quantity increases, the other increases proportionally. Be careful with inverse proportion, where one quantity increases as the other decreases, such as speed and time.

Proportional reasoning applies to direct proportion, where an increase in one quantity causes a proportional increase in another. Be aware of inverse proportion, where an increase in one quantity causes a decrease in another (e.g., more speed means less time).