Ratio, Proportion, Indices and Logarithm | CA Foundation | Ex 1 A (20)

All CA Foundation Exercises solved!
Course : 𝐂𝐀 𝐅𝐨𝐮𝐧𝐝𝐚𝐭𝐢𝐨𝐧 May, 2022 examination onwards | Topic : 𝐑𝐚𝐭𝐢𝐨, 𝐏𝐫𝐨𝐩𝐨𝐫𝐭𝐢𝐨𝐧, 𝐈𝐧𝐝𝐢𝐜𝐞𝐬 𝐚𝐧𝐝 𝐋𝐨𝐠𝐚𝐫𝐢𝐭𝐡𝐦
Exercise - 𝟏 (𝐀)
𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧 𝐍𝐮𝐦𝐛𝐞𝐫 : 𝟐𝟎 - If p : q is the sub-duplicate ratio of p - x² : q - x² then x² is …?
𝐎𝐩𝐭𝐢𝐨𝐧 (𝐀) : p / p + q
𝐎𝐩𝐭𝐢𝐨𝐧 (𝐁) : q / p + q
𝐎𝐩𝐭𝐢𝐨𝐧 (𝐂) : pq / p + q
𝐎𝐩𝐭𝐢𝐨𝐧 (𝐃) : None of these

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𝐂𝐨𝐧𝐜𝐞𝐩𝐭𝐬 𝐨𝐟 𝐑𝐚𝐭𝐢𝐨𝐬
✔️ A ratio is a comparison of the sizes of two or more quantities of the same kind by division.
✔️If p and q are two quantities of the same kind, then the fraction p/q is called the ratio of p to q.
✔️The quantities p and q are called terms of the ratio.
✔️The first term p is called antecedent.
✔️The second term q is called consequent.
✔️Both the terms of a ratio can be multiplied or divided by the same non-zero number.
✔️Usually a ratio is expressed in lowest terms.
✔️The order of the terms in a ratio is important.
✔️Ratio exists only between quantities of the same kind. Quantities to be compared must be in same units.
✔️To compare two ratios, convert them into equivalent like fractions.

𝐏𝐫𝐨𝐩𝐞𝐫𝐭𝐢𝐞𝐬 𝐨𝐟 𝐑𝐚𝐭𝐢𝐨𝐬
✔️A ratio a : b is said to be of greater inequality if a greater than b and of less inequality if a less than b.
✔️The ratio compounded of the two ratios a : b and c : d is ac : bd.
✔️The inverse ratio of a : b is b : a
✔️The duplicate ratio of a : b is a² : b²
✔️The triplicate ratio of a : b is a³ : b³
✔️The sub-duplicate ratio of a : b is √a : √b
✔️The sub-triplicate ratio of a : b is ³√a : ³√b
✔️If the ratio of two similar quantities can be expressed as a ratio of two integers, the quantities are said to be commensurable, otherwise, they are said to be incommensurable.
✔️Continued ratio is the relation or comparison between the magnitude of three or more quantities of the same kind.

𝐓𝐫𝐚𝐧𝐬𝐜𝐫𝐢𝐩𝐭 / 𝐒𝐮𝐛𝐭𝐢𝐭𝐥𝐞𝐬

We now discuss Question Number 20 from Exercise 1A. The question is... if p is to q is the sub-duplicate ratio of p minus x² is to q minus x² then x² is...? So we have a ratio p minus x² is to q minus x² and the sub duplicate ratio of this ratio is p is to q. Now in order to obtain the sub duplicate ratio of this ratio we take the square root. So root of p minus x²; that is root of antecedent and the root of consequent. It is root of q minus x² and now we have obtained the sub duplicate ratio; but this sub duplicate ratio is given to be p is to q. So the sub duplicate of p minus x² is to q minus x²... that is root of p minus x² is to root of q minus x² is p is to q and we have to check which... the value for x². Which of these is value of x²...? So let us simplify this and isolate x. So squaring both sides here the root gets eliminated. So we have p minus x² upon q minus x² and if we square the right hand side we have p² upon q². Let's cross multiply; so we have q² times p minus x² which is equal to p² times q minus x². We open up the brackets here. So we have pq² minus q²x² ...is equal to... now here... if you multiply; we have p²q minus p²x². Now we collect the x² terms on one side. Let us transfer this over here. So we have a p²x² minus q²x² is equal to.. now on the right hand side we have p²q and this term we transfer it on the right. It becomes minus pq². Now taking x² term common; so in brackets we have p² minus q². Now here we could take pq as a common factor. So in brackets we have p minus q. So from here we obtain x² as pq into p minus q divided by p² minus q². But the denominator could be further factorized...so we have x² is equal to pq into p minus q and the denominator could be split as p plus q into p minus q. Now p minus q gets cancelled. So we have x² equal to pq upon p plus q. So we have obtained x square. Let's check with the options now. pq upon p plus q. No we have pq upon p minus q over here. So the correct answer is Option D: none of these.

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𝐂𝐨𝐧𝐜𝐞𝐩𝐭𝐬
✔️A ratio is a comparison of the sizes of two or more quantities of the same kind by division.
✔️If p and q are two quantities of the same kind, then the fraction p/q is called the ratio of p to q.
✔️The quantities p and q are called terms of the ratio.
𝐏𝐫𝐨𝐩𝐞𝐫𝐭𝐢𝐞𝐬 𝐨𝐟 𝐑𝐚𝐭𝐢𝐨𝐬
✔️The sub-duplicate ratio of a : b is √a : √b