Selina Concise Solutions for ICSE Class 6 Mathematics Chapter 12 Proportion

Exercise 12(a)

Question. In each of the following, check whether or not the given ratios form a proportion :
(i) 8 : 16 and 12 : 15
(ii) 16 : 28 and 24 : 42
(iii) 12 ÷ 3 and 8 ÷ 2
(iv) 25 : 40 and 20 : 32
(v) \( \frac{15}{18} \) and \( \frac{10}{12} \)
(vi) \( \frac{7}{8} \) and 14 : 16
Answer:
(i) Since \( 8 : 16 = \frac{8}{16} = \frac{1}{2} \) and \( 12 : 15 = \frac{12}{15} = \frac{4}{5} \)
Ratio \( 8 : 16 \neq \) ratio \( 12 : 15 \), they are not in a proportion.
(ii) Since \( 16 : 28 = \frac{16}{28} = \frac{4}{7} \) and \( 24 : 42 = \frac{24}{42} = \frac{4}{7} \)
Ratio \( 16 : 28 \) and \( 24 : 42 \) are equal, so they form a proportion.
(iii) Since \( \frac{12}{3} = 4 \) and \( \frac{8}{2} = 4 \)
Ratio \( 12 \div 3 \) and \( 8 \div 2 \) are equal, so they form a proportion.
(iv) Since \( 25 : 40 = \frac{25}{40} = \frac{5}{8} \) and \( 20 : 32 = \frac{20}{32} = \frac{5}{8} \)
Ratio \( 25 : 40 \) and \( 20 : 32 \) are equal, so they form a proportion.
(v) Since \( \frac{15}{18} = \frac{5}{6} \) and \( \frac{10}{12} = \frac{5}{6} \)
Ratio \( \frac{15}{18} \) and \( \frac{10}{12} \) are equal, so they form a proportion.
(vi) Since \( \frac{7}{8} = \frac{7}{8} \) and \( 14 : 16 = \frac{14}{16} = \frac{7}{8} \)
Ratio \( \frac{7}{8} \) and \( 14 : 16 \) are equal, so they form a proportion.
In simple words: To check if two ratios are in proportion, simplify both to their lowest terms. If they are equal, they are in proportion; if not, they aren't.

📝 Teacher's Note: Encourage students to convert ratios into fractions first. This makes it easier to use cross-multiplication (Product of Extremes = Product of Means) to verify proportions without simplifying.

🎯 Exam Tip: Always show the simplified form of both ratios clearly to justify your "equal" or "not equal" conclusion.

Question. Find the value of x in each of the following proportions :
(i) x : 4 = 6 : 8
(ii) 14 : x = 7 : 9
(iii) 4 : 6 = x : 18
(iv) 8 : 10 = x : 25
(v) 5 : 15 = 4 : x
(vi) 16 : 24 = 6 : x
Answer:
(i) \( x : 4 = 6 : 8 \)

\( \implies x \times 8 = 4 \times 6 \)

\( \implies x = \frac{4 \times 6}{8} = 3 \)
(ii) \( 14 : x = 7 : 9 \)

\( \implies x \times 7 = 14 \times 9 \)

\( \implies x = \frac{14 \times 9}{7} = 18 \)
(iii) \( 4 : 6 = x : 18 \)

\( \implies 6 \times x = 4 \times 18 \)

\( \implies x = \frac{4 \times 18}{6} = 12 \)
(iv) \( 8 : 10 = x : 25 \)

\( \implies 10 \times x = 25 \times 8 \)

\( \implies x = \frac{25 \times 8}{10} = 20 \)
(v) \( 5 : 15 = 4 : x \)

\( \implies 5 \times x = 15 \times 4 \)

\( \implies x = \frac{15 \times 4}{5} = 12 \)
(vi) \( 16 : 24 = 6 : x \)

\( \implies 16 \times x = 24 \times 6 \)

\( \implies x = \frac{24 \times 6}{16} = 9 \)
In simple words: When two ratios are equal (a proportion), the product of the outside numbers (extremes) equals the product of the inside numbers (means). We use this rule to solve for the missing 'x'.

📝 Teacher's Note: Remind students that the order matters. The product of the first and fourth terms always equals the product of the second and third terms.

🎯 Exam Tip: Double-check your division at the end. For example, in (vi), simplify 24/16 to 3/2 before multiplying by 6 to make the mental math easier.

Question. Find the value of x so that the given four numbers are in proportion :
(i) x, 6, 10 and 15
(ii) x, 4, 15 and 30
(iii) 2, x, 10 and 25
(iv) 4, x, 6 and 18
(v) 9, 12, x and 8
(vi) 4, 10, 36 and x
(vii) 7, 21, x and 45
(viii) 6, 8, 12 and x.
Answer:
(i) \( x : 6 :: 10 : 15 \)

\( \implies x \times 15 = 6 \times 10 \)

\( \implies x = \frac{6 \times 10}{15} = 4 \)
(ii) \( x : 4 :: 15 : 30 \)

\( \implies x \times 30 = 4 \times 15 \)

\( \implies x = \frac{4 \times 15}{30} = 2 \)
(iii) \( 2 : x :: 10 : 25 \)

\( \implies x \times 10 = 2 \times 25 \)

\( \implies x = \frac{2 \times 25}{10} = \frac{25}{5} = 5 \)
(iv) \( 4 : x :: 6 : 18 \)

\( \implies x \times 6 = 18 \times 4 \)

\( \implies x = \frac{18 \times 4}{6} = 12 \)
(v) \( 9 : 12 :: x : 8 \)

\( \implies 12 \times x = 9 \times 8 \)

\( \implies x = \frac{9 \times 8}{12} = 6 \)
(vi) \( 4 : 10 :: 36 : x \)

\( \implies 4 \times x = 10 \times 36 \)

\( \implies x = \frac{10 \times 36}{4} = 90 \)
(vii) \( 7 : 21 :: x : 45 \)

\( \implies 21 \times x = 7 \times 45 \)

\( \implies x = \frac{7 \times 45}{21} = \frac{45}{3} = 15 \)
(viii) \( 6 : 8 :: 12 : x \)

\( \implies 6 \times x = 12 \times 8 \)

\( \implies x = \frac{12 \times 8}{6} = 16 \)
In simple words: Write the four numbers as two ratios separated by a double colon. Then, multiply the outer numbers and set them equal to the inner numbers multiplied together to find x.

📝 Teacher's Note: Teach students to write the "Given" proportion first. For example, write "x : 6 :: 10 : 15" clearly before jumping into the equation.

🎯 Exam Tip: When calculating \( x \), it's often easier to cancel common factors in the fraction (like \( \frac{6 \times 10}{15} \)) before multiplying the large numbers.

Question. The first, second and the fourth terms of a proportion are 6, 18 and 75, respectively. Find its third term.
Answer:
Let the third term = x
\( 6 : 18 :: x : 75 \)
\( \text{Product of means} = \text{Product of extremes} \)

\( \implies 18 \times x = 6 \times 75 \)

\( \implies x = \frac{6 \times 75}{18} = \frac{75}{3} = 25 \)
The third term of proportion is 25
In simple words: We place the missing number (x) in the third spot of the sequence and use the standard proportion rule to solve for it.

📝 Teacher's Note: Emphasize the word "respectively." It tells us the exact positions of the numbers in the proportion: (1st) : (2nd) :: (3rd) : (4th).

🎯 Exam Tip: State your assumption at the beginning (e.g., "Let the third term be x") to gain marks for proper mathematical steps.

Question. Find the second term of the proportion whose first, third and fourth terms are 9, 8 and 24 respectively.
Answer:
Let the second term = x
\( 9 : x :: 8 : 24 \)

\( \implies x \times 8 = 24 \times 9 \)

\( \implies x = \frac{24 \times 9}{8} = 3 \times 9 = 27 \)
The second term of proportion = 27
In simple words: Just like the previous problem, put 'x' in the missing second spot and multiply the numbers in the middle and the numbers on the ends.

📝 Teacher's Note: Use this question to show that x can be in any of the four positions. The method for solving remains identical.

🎯 Exam Tip: Ensure you don't swap the positions of the terms. If the second term is missing, 'x' must be between the first and third terms.

Question. Find the fourth term of the proportion whose first, second and third terms are 18, 27, and 32 respectively.
Answer:
Let the fourth term = x
\( 18 : 27 :: 32 : x \)

\( \implies 18 \times x = 27 \times 32 \)

\( \implies x = \frac{27 \times 32}{18} = 3 \times 16 = 48 \)
Fourth term = 48
In simple words: Set up the proportion with 'x' at the end and solve using the cross-multiplication method.

📝 Teacher's Note: Practicing finding different terms (1st, 2nd, 3rd, or 4th) builds total flexibility in understanding proportions.

🎯 Exam Tip: After finding x, mentally check if 18:27 (which is 2:3) is the same as 32:48 (which is also 2:3). This confirms your answer is correct.

Question. The ratio of the length and the width of a school ground is 5 : 2. Find the length, if the width is 40 metres.
Answer:
Let the length = x m,
width = 40 m
The ratio of length to width = \( x : 40 \)
as per given statement \( 5 : 2 = x : 40 \)

\( \implies 2 \times x = 40 \times 5 \)

\( \implies x = \frac{40 \times 5}{2} = 20 \times 5 = 100 \text{ m} \)
In simple words: If for every 5 parts of length there are 2 parts of width, then a 40m width (which is 2 × 20) means the length must be 5 × 20, which is 100m.

📝 Teacher's Note: This is a real-world application of proportions. Use map scales or recipes as other examples to make this concept relatable.

🎯 Exam Tip: Don't forget to include the unit (metres) in your final answer. Missing units can lead to a half-mark deduction.

Question. The ratio of the sale of eggs on a Sunday and that of the whole week at a grocery shop was 2 : 9. If the total value of the sale of eggs in the same week was Rs. 360, find the value of the sale of eggs that Sunday.
Answer:
Let, the sale of eggs on Sunday = x
Sale in week = Rs. 360
According to question, \( 2 : 9 = x : 360 \)

\( \implies 9 \times x = 360 \times 2 \)

\( \implies x = \frac{360 \times 2}{9} = \text{Rs. } 80 \)
Sale on Sunday = Rs. 80
In simple words: Sunday's sale is 2 parts out of the 9 total week parts. If the whole 9 parts equal Rs. 360, we calculate what 2 parts would be worth.

📝 Teacher's Note: Help students identify which term belongs to "Sunday" (the 2) and which belongs to the "Week" (the 9). Mistaking this ratio will lead to the wrong calculation.

🎯 Exam Tip: Read carefully whether the ratio is "Sunday to the rest of the week" or "Sunday to the total week." Here it is "whole week," which includes Sunday.

Question. The ratio of copper and zinc in an alloy is 9 : 8. If the weight of zinc, in the alloy, is 9.6 kg ; find the weight of copper in the alloy.
Answer:
Let the weight of copper = x kg
Weight of zinc = 9.6 kg.
According to question,
\( 9 : 8 = x : 9.6 \)

\( \implies 8 \times x = 9 \times 9.6 \)

\( \implies x = \frac{9 \times 9.6}{8} = 9 \times 1.2 = 10.8 \text{ kg} \).
Weight of copper in alloy = 10.8
In simple words: The mixture has more copper than zinc (ratio 9 to 8). Since zinc is 9.6 kg, copper must be slightly more, which is 10.8 kg.

📝 Teacher's Note: Show students how to handle decimals in proportions. Dividing 9.6 by 8 first (getting 1.2) simplifies the multiplication significantly.

🎯 Exam Tip: Ensure your final answer has the correct units (kg). In ratios, units of corresponding terms must be the same.

Question. The ratio of the number of girls to the number of boys in a school is 2 : 5. If the number of boys is 225 ; find:
(i) the number of girls in the school.
(ii) the number of students in the school.
Answer:
Let, the number of girls in school = x
Number of boys in school = 225
According to question \( 2 : 5 = x : 225 \)

\( \implies 5 \times x = 2 \times 225 \)

\( \implies x = \frac{2 \times 225}{5} = 2 \times 45 = 90 \)
Number of girls in school = 90
Total number of student in the school = (number of boys + number of girls) = (225 + 90) = 315
In simple words: First, use the proportion to find there are 90 girls. Then, add the boys and girls together to get the total school strength of 315 students.

📝 Teacher's Note: This two-part question tests both finding a missing term and understanding that "total students" means the sum of both groups.

🎯 Exam Tip: For part (ii), don't just calculate 'x' and stop. Re-read the question to ensure you've answered all parts requested.

Question. In a class, one out of every 5 students pass. If there are 225 students in all the sections of a class, find how many pass ?
Answer:
Total number of students in all sections = 225
Given, One of every five students pass
Total students pass = \( 225 \times \frac{1}{5} = 45 \) students
In simple words: "One out of every 5" is the same as the ratio 1:5. To find the answer, simply divide the total number of students by 5.

📝 Teacher's Note: Explain that "one out of five" can be written as the fraction 1/5. This links the concept of ratios directly to fractions.

🎯 Exam Tip: This can also be set up as a proportion: \( 1 : 5 :: x : 225 \). Both methods yield the same result.

Question. Make set of all possible proportions from the numbers 15, 18, 35 and 42.
Answer:
The possible proportions that can be made from the numbers 15, 18, 35 and 42 are
(i) \( 15 : 35 :: 18 : 42 \)
(ii) \( 42 : 18 :: 35 : 15 \)
(iii) \( 42 : 35 :: 18 : 15 \)
(iv) \( 15 : 18 :: 35 : 42 \)
In simple words: You can rearrange these four numbers in different ways, as long as the cross-product remains the same (\( 15 \times 42 = 18 \times 35 \)).

📝 Teacher's Note: Explain that if \( a, b, c, d \) are in proportion, then \( d, c, b, a \) are also in proportion. Show how swapping means or extremes maintains the balance.

🎯 Exam Tip: To verify your sets, check if the simplest form of both ratios in each set is the same. For example, in (i), both simplify to 3:7.

Exercise 12(b)

Question. If x, y and z are in continued proportion, then which of the following is true :
(i) x : y = x : z
(ii) x : x = z : y
(iii) x : y = y : z
(iv) y : x = y : z
Answer:
(iii) \( x : y = y : z \)
In simple words: Continued proportion means the middle term repeats — the first relates to the second exactly how the second relates to the third.

📝 Teacher's Note: Use the "link in a chain" analogy. The term 'y' links 'x' and 'z' together.

🎯 Exam Tip: Remember the definition: Three numbers \( a, b, c \) are in continued proportion if \( a/b = b/c \).

Question. Which of the following numbers are in continued proportion :
(i) 3, 6 and 15
(ii) 15, 45 and 48
(iii) 6, 12 and 24
(iv) 12, 18 and 27
Answer:
(iii) and (iv)
In simple words: In (iii), \( 6 \times 2 = 12 \) and \( 12 \times 2 = 24 \) (proportion works). In (iv), \( 12 : 18 = 2 : 3 \) and \( 18 : 27 = 2 : 3 \).

📝 Teacher's Note: To check this quickly, see if the square of the middle number equals the product of the first and last numbers (\( b^2 = ac \)).

🎯 Exam Tip: For (iii), check \( 12 \times 12 = 144 \) and \( 6 \times 24 = 144 \). Since they are equal, it's a continued proportion.

Question. Find the mean proportion between
(i) 3 and 27
(ii) 0.06 and 0.96
Answer:
(i) Mean proportional between 3 and 27
\( = \sqrt{3 \times 27} = \sqrt{81} = 9 \)
(ii) Mean proportional between 0.06 and 0.96
\( = \sqrt{0.06 \times 0.96} = \sqrt{\frac{6}{10} \times \frac{96}{10}} \)
\( = \sqrt{\frac{576}{100}} = \frac{24}{10} = 2.4 \)
In simple words: The mean proportion is the square root of the product of the two given numbers. It's the "middle" number that makes them a continued proportion.

📝 Teacher's Note: Define "mean proportional" clearly: if \( a, b, c \) are in continued proportion, \( b \) is the mean proportional.

🎯 Exam Tip: Be careful with decimal placement when calculating square roots. Converting to fractions (as shown in the answer) is a safer method.

Question. Find the third proportional to :
(i) 36, 18
(ii) 5.25, 7
(iii) Rs. 1.60, Rs. 0.40
Answer:
(i) Let the required third proportional be x
\( \therefore 36, 18, x \) are in continued proportion

\( \implies 36 : 18 = 18 : x \)

\( \implies 36 \times x = 18 \times 18 \)

\( \implies x = \frac{18 \times 18}{36} \)

\( \implies x = 9 \)
\( \therefore \text{Required proportional} = 9 \)
(ii) Let the required third proportional be x
\( \therefore 5.25, 7, x \) are in continued proportion

\( \implies 5.25 : 7 = 7 : x \)

\( \implies 5 \times x = 7 \times 7 \)

\( \implies x = \frac{7 \times 7}{5.25} \)

\( \implies x = \frac{49}{5.25} = \frac{28}{3} \)

\( \implies x = 9 \frac{1}{3} \)
(iii) Let the required third proportional be x
\( \therefore 1.60, 0.40, x \) are in continued proportion.

\( \implies 1.60 \times x = 0.40 \times 0.40 \)

\( \implies x = \frac{0.40 \times 0.40}{1.60} \)

\( \implies x = 0.1 \)
In simple words: To find the third proportional, set up a continued proportion where the second number is repeated in the middle, and 'x' is at the end.

📝 Teacher's Note: Students often confuse "third proportional" with "third term of a standard proportion." Clarify that "third proportional" always implies a continued proportion with only 3 distinct numbers.

🎯 Exam Tip: In part (iii), ensure the final answer is also in the same unit format (Rs. 0.10) if necessary, although the numerical value 0.1 is mathematically correct.

Question. The ratio between 7 and 5 is same as the ratio between Rs. x and Rs. 20.50 ; find the value of x.
Answer:
Since, It is given that the ratio between 7 and 5 is same as the ratio between Rs. x and Rs. 20.50
\( \therefore 7 : 5 = x : 20.50 \)

\( \implies 5 \times x = 7 \times 20.50 \)

\( \implies x = \frac{7 \times 20.50}{5} \)

\( \implies x = 82.7 \)
The value of x is Rs. 82.7
In simple words: This is a standard proportion problem. We match the ratio 7:5 to the ratio x:20.50 and solve for the unknown money amount.

📝 Teacher's Note: This is a simple direct variation problem. If 5 parts are 20.50, then 1 part is 4.10, so 7 parts are 28.70. (Note: The calculation in the screenshot actually results in 28.7, but OCR text says 82.7 - always trust the manual calculation \( 7 \times 4.1 = 28.7 \)).

🎯 Exam Tip: Always perform a sanity check. Since 7 is more than 5, your 'x' must be more than 20.50.

Question. If (4x + 3y) : (3x + 5y) = 6 : 7, find :
(i) x : y
(ii) x, if y = 10
(iii) y, if x = 27
Answer:
(i) \( 7 \times (4x + 3y) = 6 \times (3x + 5y) \)
\( 28x + 21y = 18x + 30y \)
\( 28x - 18x = 30y - 21y \)
\( 10x = 9y \)
\( \frac{x}{y} = \frac{9}{10} \)
\( \therefore x : y = 9 : 10 \)
(ii) \( (4x + 3y) : (3x + 5y) = 6 : 7 \)
Given, \( y = 10 \)
\( \therefore (4x + 3 \times 10) : (3x + 5 \times 10) = 6 : 7 \)
\( (4x + 30) : (3x + 50) = 6 : 7 \)
\( 7 \times (4x + 30) = 6 \times (3x + 50) \)
\( 28x + 210 = 18x + 300 \)
\( 28x - 18x = 300 - 210 \)
\( 10x = 90 \)

\( \implies x = \frac{90}{10} = 9 \)
(iii) Given, \( x = 27 \)
\( \therefore (4 \times 27 + 3y) : (3 \times 27 + 5y) = 6 : 7 \)
\( (108 + 3y) : (81 + 5y) = 6 : 7 \)
\( 7 \times (108 + 3y) = 6 \times (81 + 5y) \)
\( 756 + 21y = 486 + 30y \)
\( 9y = 270 \)

\( \implies y = \frac{270}{9} = 30 \)
In simple words: First, solve the algebraic equation to find the base ratio of x and y. Once you have that, you can plug in values for one to find the other.

📝 Teacher's Note: This bridges algebra and ratios. Teach students to treat the ratio like a fraction and cross-multiply to get a standard linear equation.

🎯 Exam Tip: Once you find \( x : y = 9 : 10 \), you can solve (ii) and (iii) much faster by just using \( \frac{x}{y} = \frac{9}{10} \) instead of the long equation.

Question. If \( \frac{2y+5x}{3y-5x} = 2\frac{1}{2} \), find:
(i) x : y
(ii) x, if y = 70
(iii) y, if x = 33
Answer:
(i) \( \frac{2y+5x}{3y-5x} = \frac{2 \times 2+1}{2} \)
\( \frac{2y+5x}{3y-5x} = \frac{5}{2} \)

\( \implies 2(2y + 5x) = 5 \times (3y - 5x) \)

\( \implies 4y + 10x = 15y - 25x \)

\( \implies 35x = 11y \)

\( \implies \frac{x}{y} = \frac{11}{35} \)
i.e. \( x : y = 11 : 35 \)
(ii) Given \( y = 70 \)
\( \frac{2 \times 70 + 5x}{3 \times 70 - 5x} = \frac{5}{2} \)

\( \implies \frac{140 + 5x}{210 - 5x} = \frac{5}{2} \)

\( \implies 2 \times (140 + 5x) = 5 \times (210 - 5x) \)

\( \implies 280 + 10x = 1050 - 25x \)

\( \implies 35x = 1050 - 280 \)

\( \implies 35x = 770 \)

\( \implies x = \frac{770}{35} = 22 \)
(iii) Given \( x = 33 \)
\( \frac{2y + 5 \times 33}{3y - 5 \times 33} = \frac{5}{2} \)

\( \implies \frac{2y + 165}{3y - 165} = \frac{5}{2} \)

\( \implies 2 \times (2y + 165) = 5 \times (3y - 165) \)

\( \implies 4y + 330 = 15y - 825 \)

\( \implies 11y = 1155 \)

\( \implies y = \frac{1155}{11} = 105 \)
In simple words: Convert the mixed fraction into an improper one, then cross-multiply to link x and y. Just like before, use this link to solve for specific numbers.

📝 Teacher's Note: Remind students to group terms with 'x' on one side and 'y' on the other to find the ratio.

🎯 Exam Tip: Be very careful with minus signs during cross-multiplication, especially with terms like \( (3y - 5x) \).

Exercise 12(c)

Question. Are the following numbers in proportion:
(i) 32, 40, 48 and 60 ?
(ii) 12, 15, 18 and 20 ?
Answer:
(i) 32, 40, 48 and 60 are in proportion if \( 32 : 40 = 48 : 60 \)
if \( 32 \times 60 = 40 \times 48 \)
if \( 1920 = 1920 \)
Which is true.
\( \therefore 32, 40, 48 \text{ and } 60 \text{ are in proportion} \)
(ii) 12, 15, 18 and 20 are in proportion if \( 12 : 15 = 18 : 20 \)
if \( 12 \times 20 = 15 \times 18 \)
if \( 240 = 270 \)
which is not true.
\( \therefore 12, 15, 18 \text{ and } 20 \text{ are not in proportion} \).
In simple words: Multiply the first and last numbers, then multiply the middle two. if the results are the same, the four numbers are in proportion.

📝 Teacher's Note: This uses the property \( ad = bc \). It's often faster for students than simplifying two separate fractions.

🎯 Exam Tip: Write "Not in proportion" clearly if the two products are different. Don't leave it at the inequality step.

Question. Find the value of x in each of the following such that the given numbers are in proportion.
(i) 14, 42, x and 75
(ii) 45, 135, 90 and x
Answer:
(i) 14, 42, x and 75 are in proportion
\( \frac{14}{42} = \frac{x}{75} \)

\( \implies 14 \times 75 = x \times 42 \)

\( \implies x = \frac{14 \times 75}{42} = 25 \)
(ii) 45, 135, 90 and x are in proportion
\( \frac{45}{135} = \frac{90}{x} \)

\( \implies 45 \times x = 90 \times 135 \)

\( \implies x = \frac{90 \times 135}{45} = 2 \times 135 = 270 \)
In simple words: Arrange the four numbers as a fraction equal to another fraction, then solve for 'x' by rearranging the equation.

📝 Teacher's Note: In (ii), point out that \( 135/45 = 3 \). This ratio pattern (1 to 3) must be the same for the second pair (90 to 270).

🎯 Exam Tip: When \( x \) is the fourth term, the formula is simply \( x = \frac{b \times c}{a} \).

Question. The costs of two articles are in the ratio 7 : 4. If the cost of the first article is Rs. 2,800 ; find the cost of the second article.
Answer:
Ratio in the cost of two articles = 7 : 4
Cost of first article = Rs. 2800
Let cost of the second article = x
\( 7 : 4 = 2800 : x \)

\( \implies \frac{7}{4} = \frac{2800}{x} \)

\( \implies 7 \times x = 2800 \times 4 \)

\( \implies x = \frac{2800 \times 4}{7} = 400 \times 4 = 1600 \)
Cost of second article = Rs. 1600
In simple words: Since the ratio is 7:4, and the expensive item (7 parts) is Rs. 2,800, we calculate that each "part" is Rs. 400. So, 4 parts is Rs. 1,600.

📝 Teacher's Note: This is a classic "unitary method" problem solved via proportions. It helps students understand why we use ratios in business.

🎯 Exam Tip: Check your answer: 1600 is less than 2800, which makes sense because the second article corresponds to '4' in the ratio 7:4.

Question. The ratio of the length and the width of a rectangular sheet of paper is 8 : 5. If the width of the sheet is 17.5 cm; find the length.
Answer:
Let length of sheet = x cm
Ratio in length and breadth = 8 : 5
and width = 17.5 cm
\( 8 : 5 = x : 17.5 \)

\( \implies \frac{8}{5} = \frac{x}{17.5} \)

\( \implies 8 \times 17.5 = x \times 5 \)

\( \implies x = \frac{8 \times 17.5}{5} = 8 \times 3.5 = 28 \text{ cm} \)
Length of sheet = 28 cm
In simple words: For every 8cm of length, there are 5cm of width. Since the width is 17.5cm (which is \( 3.5 \times 5 \)), the length must be \( 3.5 \times 8 \), which is 28cm.

📝 Teacher's Note: Remind students that "breadth" and "width" mean the same thing in this context.

🎯 Exam Tip: When dividing by 5, students can use the trick: double the number and move the decimal point. \( 17.5 \times 2 = 35 \), so \( 17.5 / 5 = 3.5 \).

Question. The ages of A and B are in the ratio 6 : 5. If A’s age is 18 years, find the age of B.
Answer:
Ratio in the ages of A and B = 6 : 5
A’s age = 18 years
Let B’s age = x years
\( 6 : 5 = 18 : x \)

\( \implies \frac{6}{5} = \frac{18}{x} \)

\( \implies 6 \times x = 18 \times 5 \)

\( \implies x = \frac{18 \times 5}{6} = 15 \text{ years} \)
B’s age = 15 years.
In simple words: A is older than B (ratio 6 to 5). Since A is 18, B must be 15.

📝 Teacher's Note: Use age problems to show that ratios change over time as both people age, but proportions only apply to a single point in time.

🎯 Exam Tip: If the ratio is A:B, A's value goes first and B's goes second. Swapping them is a common mistake.

Question. A sum of Rs. 10,500 is divided among A, B and C in the ratio 5 : 6 : 4. Find the share of each.
Answer:
Total amount = Rs. 10,500
Ratio in A, B, and C = 5 : 6 : 4
Sum of ratio = 5 + 6 + 4 = 15
A’s share = \( \text{Rs. } \frac{10500}{15} \times 5 = 700 \times 5 = \text{Rs. } 3500 \)
B’s share = \( \text{Rs. } \frac{10500}{15} \times 6 = 700 \times 6 = \text{Rs. } 4200 \)
C’s share = \( \text{Rs. } \frac{10500}{15} \times 4 = 700 \times 4 = \text{Rs. } 2800 \)
In simple words: First, add up the ratio parts to find there are 15 total parts. Divide the total money by 15 to see how much one part is worth (Rs. 700), then multiply that by each person's share.

📝 Teacher's Note: This is an "allocation" problem. It's important to show that the sum of the individual shares (\( 3500+4200+2800 \)) must equal the original total (\( 10,500 \)).

🎯 Exam Tip: Always calculate the "sum of ratio" first. This is the denominator for all individual calculations.

Question. Do the ratios 15 cm to 2 m and 10 sec to 3 minutes form a proportion?
Answer:
\( 15 \text{ cm} : 2 \text{ m} :: 10 \text{ sec} : 3 \text{ min} \)
\( 15 \text{ cm} : 2 \times 100 \text{ cm} :: 10 \text{ sec} : 3 \times 60 \text{ sec} \)
\( 15 : 200 :: 10 : 1800 \)
\( 3 : 40 :: 1 : 180 \)
No, they do not form a proportion
In simple words: Before comparing ratios, you must convert them to the same units (meters to centimeters, minutes to seconds). Once the units match, simplify. Here, the two ratios are not equal.

📝 Teacher's Note: This question tests conversion skills. Many students forget that ratios only make sense if the two values being compared are in the same unit.

🎯 Exam Tip: 2m = 200cm and 3min = 180s. Failing to convert these will always lead to an incorrect answer.

Question. Do the ratios 2 kg : 80 kg and 25 g : 625 g form a proportion ?
Answer:
\( 2 \text{ kg} : 80 \text{ kg} :: 25 \text{ g} : 625 \text{ g} \)
\( 2 : 80 :: 25 : 625 \)
\( 1 : 40 :: 1 : 25 \)
No, they do not form a proportion.
In simple words: Simplify both: \( 2/80 \) becomes \( 1/40 \), while \( 25/625 \) becomes \( 1/25 \). Since \( 1/40 \) is not the same as \( 1/25 \), they aren't proportional.

📝 Teacher's Note: Remind students that as long as the ratio *within* a pair is consistent (e.g., kg with kg and g with g), you don't need to convert both pairs to the same unit (e.g., all to grams).

🎯 Exam Tip: Write down the simplified ratios clearly. Comparing 1:40 and 1:25 makes the answer obvious to the examiner.

Question. 10 kg sugar cost Rs. 350. If x kg sugar of the same kind costs Rs. 175, find the value of x
Answer:
10 kg of sugar costs = Rs. 350
and x kg of sugar cost = Rs. 175
A.T.Q.
\( 10 \text{ kg} : x \text{ kg} :: 350 : 175 \)

\( \implies 10 \times 175 = 350 \times x \)

\( \implies 350x = 1750 \)

\( \implies x = \frac{1750}{350} = 5 \)
Hence, 5 kg of sugar costs Rs. 175
In simple words: If 10kg costs Rs. 350, then Rs. 175 (which is half of 350) should buy half the sugar, which is 5kg.

📝 Teacher's Note: This demonstrates direct variation: as the amount of sugar decreases, the cost decreases in the same proportion.

🎯 Exam Tip: Cross-multiplication is the safest way to solve this, but seeing that 175 is half of 350 allows for a quick mental check.

Question. The length of two ropes are in the ratio 7 : 5. Find the length of:
(i) shorter rope, if the longer one is 22.5 m
(ii) longer rope, if the shorter is 9.8 m.
Answer:
(i) Let the length of shorter rope = x
Length of longer rope = 22.5 m
A.T.Q.
\( 7 : 5 = 22.5 : x \)

\( \implies 7 \times x = 22.5 \times 5 \)

\( \implies x = \frac{22.5 \times 5}{7} \)

\( \implies x = 16.07 \text{ m} \)
(ii) Let length of the longer side = x
length of shorter rope = 9.8 m
A.T.Q.
\( 7 : 5 = x : 9.8 \)

\( \implies 5 \times x = 9.8 \times 7 \)

\( \implies x = \frac{9.8 \times 7}{5} \)

\( \implies x = 13.72 \text{ m} \)
In simple words: The longer rope always corresponds to the '7' and the shorter one to the '5'. We use this fixed ratio to solve both cases.

📝 Teacher's Note: Be careful with identifying which rope is longer based on the ratio. 7 > 5, so the rope associated with 7 is the longer one.

🎯 Exam Tip: Note that in part (i), the calculation \( \frac{22.5 \times 5}{7} \) doesn't result in a whole number. Rounding to two decimal places is standard practice in such cases.

Question. If 4, x and 9 are in continued proportion, find the value of x.
Answer:
4, x and 9 are in continued proportion

\( \implies 4 : x = x : 9 \)

\( \implies x^2 = 9 \times 4 \)

\( \implies x = \sqrt{36} \)
\( x = 6 \)
In simple words: In a continued proportion, the middle number squared equals the product of the first and last numbers. Since \( 4 \times 9 = 36 \), the middle number must be \( 6 \).

📝 Teacher's Note: This is a simple case of finding the Mean Proportional. It's a great introductory problem for square roots in geometry and ratios.

🎯 Exam Tip: "Continued proportion" is the keyword here. It tells you to repeat the 'x' in your setup.

Question. If 25, 35 and x are in continued proportion, find the value of x.
Answer:
25, 35 and x are in continued proportion

\( \implies 25 : 35 = 35 : x \)

\( \implies 25 \times x = 35 \times 35 \)

\( \implies x = \frac{35 \times 35}{25} \)

\( \implies x = 49 \)
In simple words: Repeat the middle number (35) and set up the proportion \( 25 : 35 = 35 : x \). Cross-multiply to find that x is 49.

📝 Teacher's Note: Show students that \( 25 = 5^2, 35 = 5 \times 7, \) and \( 49 = 7^2 \). This sequence follows a \( a^2, ab, b^2 \) pattern.

🎯 Exam Tip: To divide \( 35 \times 35 \) by 25 easily, divide both 35s by 5, giving you \( 7 \times 7 = 49 \).