Fom 1 Lecture 4

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JEE Mathematics: Ratio, Proportion, k-method, Interval (Lecture 04)

Topic 1: Ratio, Proportion and k-method

  • A ratio is a comparison of two quantities of the same kind, expressed as $a:b$ or $\frac{a}{b}$.
  • A proportion is an equation stating that two ratios are equal, i.e., $\frac{a}{b} = \frac{c}{d}$.
  • k-method: If a series of ratios are equal, such as $\frac{a}{b} = \frac{c}{d} = k$, then we can express the numerators in terms of denominators as $a = bk$ and $c = dk$.
  • To combine two separate ratios like $a:b$ and $b:c$, find the Least Common Multiple (LCM) of the numerical values of the common variable $b$.
  • ગુણોત્તર (Ratio) એ સમાન પ્રકારની બે માત્રાઓની સરખામણી છે, જેને $a:b$ અથવા $\frac{a}{b}$ તરીકે દર્શાવવામાં આવે છે.
  • પ્રમાણ (Proportion) એ એક સમીકરણ છે જે દર્શાવે છે કે બે ગુણોત્તર સમાન છે, એટલે કે $\frac{a}{b} = \frac{c}{d}$.
  • k-method: જો ગુણોત્તરો સમાન હોય, જેમ કે $\frac{a}{b} = \frac{c}{d} = k$, તો આપણે અંશને છેદના સ્વરૂપમાં $a = bk$ અને $c = dk$ તરીકે દર્શાવી શકીએ છીએ.
  • $a:b$ અને $b:c$ જેવા બે અલગ ગુણોત્તરને ભેગા કરવા માટે, સામાન્ય ચલ $b$ ની આંકડાકીય કિંમતોનો લ.સા.અ. (LCM) શોધો.
Extra Illustration

If $x:y = 2:5$, then find the value of $\frac{3x + 2y}{4x – y}$.

જો $x:y = 2:5$ હોય, તો $\frac{3x + 2y}{4x – y}$ ની કિંમત શોધો.

Solution:

Given $\frac{x}{y} = \frac{2}{5}$. Let $x = 2k$ and $y = 5k$, where $k$ is a constant.
Substitute these values into the given expression:

$$ \frac{3(2k) + 2(5k)}{4(2k) – (5k)} $$
$$ \frac{6k + 10k}{8k – 5k} = \frac{16k}{3k} $$

The $k$ cancels out, leaving:

$$ \frac{16}{3} $$

Classwork Question 1

If $x:y=3:4$ then find $7x-4y : 3x+y$.

જો $x: y=3: 4$ તો $7x-4y : 3x + y$ મેળવો.

Solution:

Given the ratio $\frac{x}{y} = \frac{3}{4}$, we can assume $x = 3k$ and $y = 4k$.

We need to find the ratio $\frac{7x – 4y}{3x + y}$. Substitute the values of $x$ and $y$:

$$ \frac{7(3k) – 4(4k)}{3(3k) + (4k)} $$
$$ \frac{21k – 16k}{9k + 4k} $$
$$ \frac{5k}{13k} $$

Cancel the common constant $k$:

$$ \frac{5}{13} $$

So, the ratio is $5:13$.

Classwork Question 2

If $\frac{a}{b}=\frac{2}{3}, \frac{b}{c}=\frac{4}{5}$ then find the value of $\frac{a+b}{b+c}$.

જો $\frac{a}{b}=\frac{2}{3}, \frac{b}{c}=\frac{4}{5}$ તો $\frac{a+b}{b+c} = \_$.

Solution:

We are given two separate ratios involving $b$. To combine them, we need to make the corresponding value of $b$ the same in both fractions by finding the LCM.
In $\frac{a}{b} = \frac{2}{3}$, the value corresponding to $b$ is $3$.
In $\frac{b}{c} = \frac{4}{5}$, the value corresponding to $b$ is $4$.

The LCM of $3$ and $4$ is $12$. Multiply the numerators and denominators to make $b = 12$:

$$ \frac{a}{b} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} $$
$$ \frac{b}{c} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15} $$

Now we have a continuous ratio $a:b:c = 8:12:15$.
Let $a = 8k$, $b = 12k$, and $c = 15k$. Substitute these into the target expression $\frac{a+b}{b+c}$:

$$ \frac{8k + 12k}{12k + 15k} = \frac{20k}{27k} $$
$$ \frac{20}{27} $$

Linked HW: HW Q1, HW Q2, HW Q3

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