Key Concept: Ratio Simplification, Proportional Reasoning

c) 3:2
[Solution Description]
To simplify the ratio 9:6, we divide both numbers by their greatest common divisor (GCD).
1. GCD of 9 and 6 is 3.
2. Divide both terms by 3: $\frac{9}{3} : \frac{6}{3} = 3:2$.
The simplified ratio is 3:2.

Key Concept: Proportional Reasoning, Similarity in Change

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that images A, C, and D appear similar due to proportional scaling of their dimensions, which is true as seen from the table (width and height scaled by $\frac{1}{2}$ for C and $1.5$ for D relative to A). The reason correctly explains this by stating that similarity requires equal scaling factors for both dimensions. Both statements are true, and the reason accurately justifies the assertion.

Key Concept: Scaling Factors

b) 1.5
[Solution Description]
To find the scaling factor:
Scaling factor = $\frac{\text{New Width}}{\text{Original Width}} = \frac{120}{80} = 1.5$.
Since the images are similar, the height also scales by the same factor.
New Height = $50 \times 1.5 = 75$ mm.

Key Concept: Simplified Ratio

c) $3:2$
[Solution Description]
To simplify the ratio $30:20$, divide both numbers by their HCF (10):
$30 / 10 : 20 / 10 = 3:2$
The simplified ratio is $3:2$.

Key Concept: Verification of Similarity

b) $2 : 1$
[Solution Description]
The ratio of width to height for Image B is $40 : 20$. Dividing both terms by their HCF (20), we get:
$\frac{40}{20} : \frac{20}{20} = 2 : 1$
The simplest form of the ratio for Image B is $2 : 1$.

Key Concept: Width-Height Ratio

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine similarity between images, we compare their width-to-height ratios in simplest form. For image A, the ratio is $60:40$, which simplifies to $3:2$. Similarly, for image C, the ratio is $30:20$, simplifying to $3:2$, and for image D, the ratio is $90:60$, also simplifying to $3:2$. Since all three ratios reduce to the same simplest form, the images are proportionally similar. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Multiplicative vs. additive changes, Proportionality

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that images A, C, and D remain similar because their dimensions change multiplicatively by the same factor. This is true because the ratio of width to height for these images is consistently $3:2$, indicating proportional scaling. The Reason states that multiplicative scaling preserves the ratio of width to height, while additive scaling does not. This is also true and correctly explains why the images remain similar under multiplicative scaling but not additive scaling. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Proportional Scaling Factor

d) 2
[Solution Description]
To find the scaling factor, divide the dimensions of Image Y by those of Image X:
Width factor = $\frac{100}{50} = 2$,
Height factor = $\frac{60}{30} = 2$.
Since both factors are equal, the scaling factor is 2.

Key Concept: Ratios, Simplifying Ratios

c) 21
[Solution Description]
Given the ratio of boys to girls is $5 : 3$ and there are 35 boys.
Let the number of girls be $x$.
According to the ratio:
$5 : 3 = 35 : x$
This can be written as:
$\frac{5}{3} = \frac{35}{x}$
Cross-multiplying:
$5x = 105$
Solving for $x$:
$x = \frac{105}{5} = 21$
Therefore, there are 21 girls.

Key Concept: Proportional Ratios

c) $12 : 18$
[Solution Description]
A ratio proportional to $4 : 6$ must simplify to the same form or satisfy $ad = bc$. The simplest form of $4 : 6$ is $2 : 3$. Check each option:
– $8 : 12$ simplifies to $2 : 3$ (proportional).
– $6 : 9$ simplifies to $2 : 3$ (proportional).
– $10 : 15$ simplifies to $2 : 3$ (proportional).
– $12 : 18$ simplifies to $2 : 3$ (proportional).
Among the given options, $12 : 18$ is correct.

Key Concept: Ratio Simplification

b) $1 : 3$
[Solution Description]
To simplify the ratio $15 : 45$, find the highest common factor (HCF) of 15 and 45, which is 15. Divide both terms by 15 to get $1 : 3$.

Key Concept: Proportional Ratios

b) $16 : 24$
[Solution Description]
First, simplify the given ratio $12 : 18$ by dividing both terms by their HCF (6):
$\frac{12}{6} : \frac{18}{6} = 2 : 3$
Now check the options for ratios that simplify to $2 : 3$:
$16 : 24 \implies \frac{16}{8} : \frac{24}{8} = 2 : 3$
Thus, $16 : 24$ is proportional to $12 : 18$.

Key Concept: Simplest Form of Ratio

c) $2 : 3$
[Solution Description]
To simplify the ratio $60 : 90$, divide both terms by their HCF. The HCF of 60 and 90 is 30. Dividing both by 30 gives $2 : 3$.

Key Concept: Proportional Ratios

a) $8 : 12$
[Solution Description]
To determine which ratio is proportional to $4 : 6$, first simplify $4 : 6$ to its simplest form. The HCF of 4 and 6 is 2. Dividing both terms by 2 gives:
$\frac{4}{2} : \frac{6}{2} = 2 : 3$
Now, check each option by simplifying it to see if it matches $2 : 3$.
Option a) simplifies to $8 : 12$ → $\frac{8}{4} : \frac{12}{4} = 2 : 3$. This matches the simplified form of $4 : 6$.
The other options do not simplify to $2 : 3$.

Key Concept: Interpretation of Ratios

c) 15 cups
[Solution Description]
The given ratio of sugar to flour is $3 : 5$. For every 3 cups of sugar, 5 cups of flour are needed. If 9 cups of sugar are used, we need to find the corresponding amount of flour.
First, determine the scaling factor:
$\text{Scaling factor} = \frac{9}{3} = 3$
Multiply the flour part of the ratio by this scaling factor:
$5 \times 3 = 15$
Thus, 15 cups of flour are needed.

Key Concept: Simplest Form of Ratios

b) $3 : 2$
[Solution Description]
To find the simplest form of the ratio $90 : 60$, divide both terms by their Highest Common Factor (HCF). The HCF of 90 and 60 is 30. Dividing both terms by 30 gives:
$\frac{90}{30} : \frac{60}{30} = 3 : 2$
Thus, the simplest form of the ratio $90 : 60$ is $3 : 2$.

Key Concept: Proportional Ratios

d) $25 : 18$
[Solution Description]
To determine if ratios are proportional, we simplify them to their simplest form. The given ratio is $5:3$. For another ratio to be proportional, it must simplify to the same ratio ($5:3$) when reduced.
Let’s check each option:
\begin{itemize}

Key Concept: Proportional Ratios

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, let’s verify the assertion by simplifying both ratios:
– For $12:18$, dividing both terms by 6 (HCF) gives $2:3$.
– For $8:12$, dividing both terms by 4 (HCF) gives $2:3$.
Since both simplified ratios are equal ($2:3$), the assertion is true. The reason correctly explains why the assertion is true because it shows that both ratios simplify to the same simplest form.

Key Concept: Proportional Ratios, Simplest Form

a) Yes, because both simplify to $4:3$
[Solution Description] First, simplify both ratios to their simplest form. For the first ratio $48:36$, the HCF of 48 and 36 is 12. Dividing both terms by 12 gives $\frac{48}{12} : \frac{36}{12} = 4:3$. For the second ratio $64:48$, the HCF of 64 and 48 is 16. Dividing both terms by 16 gives $\frac{64}{16} : \frac{48}{16} = 4:3$. Since both simplified ratios are equal ($4:3$), the original ratios are proportional.

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check if the ratios are proportional, we simplify them:
1. Simplifying $60:40$ by dividing both terms by their HCF (20): $60 \div 20 : 40 \div 20 = 3:2$.
2. Simplifying $90:60$ by dividing both terms by their HCF (30): $90 \div 30 : 60 \div 30 = 3:2$.
Since both simplified forms are equal ($3:2$), the Assertion is true. The Reason correctly explains that they are proportional because their simplest forms match.

Key Concept: Simplifying Ratios

b) $2 : 3$
[Solution Description]
To simplify the ratio $60 : 90$, divide both terms by their HCF. The HCF of 60 and 90 is 30. Dividing both terms by 30 gives $\frac{60}{30} : \frac{90}{30} = 2 : 3$. Thus, the simplified ratio is $2 : 3$.

Key Concept: Simplifying ratios, Proportional reasoning

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let the side lengths of the two cubes be $3k$ and $5k$ respectively. The surface area of a cube is given by $6 \times (\text{side length})^2$. So, the surface areas are:
Cube 1: $6 \times (3k)^2 = 6 \times 9k^2 = 54k^2$
Cube 2: $6 \times (5k)^2 = 6 \times 25k^2 = 150k^2$
The ratio of their surface areas is $54k^2 : 150k^2$, which simplifies to $9:25$ when divided by their HCF of 6$k^2$. The reason correctly explains the assertion since the surface area indeed depends on the square of the side length.

Key Concept: Proportional Ratios

d) $5 : 7$
[Solution Description]
First, simplify $8 : 12$ by dividing both terms by their HCF (which is 4), giving $2 : 3$. Now check each option:
\begin{itemize}
– Option a) $4 : 6$ simplifies to $2 : 3$ (proportional).
– Option b) $10 : 15$ simplifies to $2 : 3$ (proportional).
– Option c) $6 : 9$ simplifies to $2 : 3$ (proportional).
– Option d) $5 : 7$ does not simplify to $2 : 3$ (not proportional).
\end{itemize}
All except option d) are proportional to $8 : 12$.

Key Concept: Problem solving with proportional reasoning, Unit conversions

a) 304,920 sq ft
[Solution Description]
Given that 3 acres = 130,680 sq ft, then 1 acre = $\frac{130,680}{3} = 43,560$ sq ft. For 7 acres: $7 \times 43,560 = 304,920$ sq ft.

Key Concept: Assertion and Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the ratio $60:40$ simplifies to $3:2$, which is correct because dividing both terms by their HCF (20) gives $\frac{60}{20}:\frac{40}{20} = 3:2$. The Reason correctly explains that the HCF of 60 and 40 is 20, which is indeed used to simplify the ratio. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Age ratios, Changing ratios

c) 3 : 4
[Solution Description]
At Rahul’s age 6, sister’s age is $6 \times 2 = 12$ years.
Age difference is always $12 – 6 = 6$ years.
When Rahul is 18, sister will be $18 + 6 = 24$ years.
The ratio will be $18 : 24$.
Simplifying by dividing both terms by 6 gives $3 : 4$.

Key Concept: Finding Missing Terms in Proportional Ratios

c) $24 : 36$
[Solution Description]
The given ratio $8 : 12$ can be simplified to $2 : 3$. To find the missing term in $24 : \_\_\_\_$, observe that 24 is obtained by multiplying the first term (8) by 3. Therefore, multiply the second term (12) by 3:
$12 \times 3 = 36$
So, the proportional ratio is $24 : 36$.

Key Concept: Ratios in Their Simplest Form, Proportional Ratios

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine whether the assertion and reason are correct:
1. Simplify the ratio $14 : 21$ by dividing both terms by their HCF, which is 7. This gives the simplified form as $2 : 3$.
2. Similarly, simplify the ratio $6 : 9$ by dividing both terms by their HCF, which is 3. This also gives the simplified form as $2 : 3$.
3. Since both simplified forms are equal ($2 : 3$), the ratios $14 : 21$ and $6 : 9$ are indeed proportional.
4. The Reason states that two ratios are proportional if their terms change by the same multiplicative factor. This is true, as seen here where both $14 : 21$ and $6 : 9$ reduce to the same simplest form. Hence, the Reason correctly explains the Assertion.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Simplest Form Calculation

c) $2 : 3$
[Solution Description]
Find the HCF of 72 and 108, which is 36. Divide both terms by 36:
$72 \div 36 = 2 \\ \\ 108 \div 36 = 3$
The simplified ratio is $2 : 3$.

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To simplify the ratio $12 : 18$, we find the HCF of 12 and 18, which is 6. Dividing both terms by 6 gives $\frac{12}{6} : \frac{18}{6} = 2 : 3$. Therefore, the Assertion is true. The Reason states that the HCF of 12 and 18 is 6, which is also true and correctly explains why the ratio simplifies to $2 : 3$.

Key Concept: Proportional ratios, Cross-multiplication

a) $10 : 16$
[Solution Description]
To check proportionality, we can cross-multiply or simplify the ratios. For two ratios to be proportional, their simplest forms must be equal. The simplest form of $5 : 8$ is already $5 : 8$. Now, let’s check each option:
– $10 : 16$ simplifies to $5 : 8$ (HCF = 2).
Thus, $10 : 16$ is proportional to $5 : 8$.

Key Concept: Simplifying Ratios Using HCF

a)
[Solution Description]
To simplify the ratio $84:56$, we first find the HCF of 84 and 56.
Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
The highest common factor is 28. Dividing both terms by 28 gives:
$84 \div 28 = 3$
$56 \div 28 = 2$
So the simplified form is $3:2$. Both Assertion and Reason are true, and Reason correctly explains how to simplify the ratio using HCF.

Key Concept: Equivalence of ratios in simplest form

c) 12
[Solution Description]
First, simplify the ratio $48 : 64$ to its simplest form. The HCF of 48 and 64 is 16. Dividing both terms by 16 gives $3 : 4$. Since $48 : 64 :: 9 : x$, their simplest forms must be equal. So, $3 : 4 = 9 : x$. Cross-multiplying gives $3x = 36$, so $x = 12$.

Key Concept: Missing Term in Proportions

b) 28
[Solution Description]
Since the ratios are proportional, the product of means equals the product of extremes:
$7 \times 48 = 12 \times x \implies 336 = 12x \implies x = \frac{336}{12} = 28$
So, $x = 28$.

Key Concept: Ratios in Simplest Form

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check if the ratios are proportional, we need to simplify both ratios:
1. For $18:12$, divide both terms by HCF of 18 and 12 which is 6. This gives $3:2$.
2. For $27:18$, divide both terms by HCF of 27 and 18 which is 9. This gives $3:2$.
Since both simplified ratios are equal ($3:2$), they are proportional. Also, the Reason correctly explains the condition for proportionality of ratios as it states that two ratios are proportional if their simplest forms are equal.

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check if the ratios are proportional, simplify both ratios to their simplest form:
\begin{itemize}

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check if the ratios are proportional, we simplify both ratios to their simplest form.
Step 1: Simplify $6:4$. The HCF of 6 and 4 is 2. Dividing both terms by 2 gives $3:2$.
Step 2: Simplify $9:6$. The HCF of 9 and 6 is 3. Dividing both terms by 3 also gives $3:2$.
Since both simplified ratios are equal ($3:2$), they are proportional. Additionally, the Reason correctly explains why the Assertion is true.

Key Concept: Simplifying Ratios, Proportionality

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine if the assertion is true or false:
1. Simplify $36:48$ by dividing both terms by their HCF. The HCF of 36 and 48 is 12. Simplifying gives $3:4$.
2. Simplify $27:36$ by dividing both terms by their HCF. The HCF of 27 and 36 is 9. Simplifying gives $3:4$.
3. Since both simplified ratios are equal ($3:4$), they are proportional, making the Assertion true.
4. The Reason states that two ratios are proportional if their simplest forms are equal, which is also true and correctly explains why the Assertion is true.

Key Concept: Proportional Reasoning

c) 40
[Solution Description]
Using cross-multiplication for proportion $5:8 :: 25:x$,
$5 \times x = 8 \times 25$.
Solving for $x$:
$5x = 200$
$x = 200 ÷ 5 = 40$.

Key Concept: Simplification and Proportionality

c) The simplified form is $2:3$ and proportional to $2:3$
[Solution Description]
To simplify $24:36$, divide both terms by their HCF (12).
$24 ÷ 12 = 2$ and $36 ÷ 12 = 3$.
So, the simplified form is $2:3$.
Since the given ratio is also $2:3$, they are proportional.

Key Concept: Ratio Application

b) 6 cups
[Solution Description]
The ratio of flour to water is 4:6 or simplified as 2:3. To find the required flour for 9 cups of water:
$\frac{2}{3} = \frac{x}{9} \\ \\ 3x = 18 \\ \\ x = 6$
You need 6 cups of flour.

Key Concept: Proportional Division

a) \Rs.1,800 and \Rs.2,700
[Solution Description]
To divide \Rs.4,500 in the ratio $2 : 3$, we first find the total parts: $2 + 3 = 5$. Then, the share for the first part is $\frac{2}{5} \times 4500 = 1800$ and for the second part is $\frac{3}{5} \times 4500 = 2700$.

Key Concept: Compound ratios, proportional reasoning

c) 1:1
[Solution Description]
Calculate worker-days for each wall:
Wall X: 4 workers × 15 days = 60 worker-days
Wall Y: 6 workers × 10 days = 60 worker-days
Since both walls require same worker-days, their efficiencies are equal.
Ratio is therefore 1:1.

Key Concept: Proportional Reasoning

c) 600 km
[Solution Description]
To find the distance traveled in 10 hours at the same speed:
$\text{Speed} = \frac{360 \, \text{km}}{6 \, \text{hours}} = 60 \, \text{km/h} \\ \\ \text{Distance} = 60 \, \text{km/h} \times 10 \, \text{hours} = 600 \, \text{km}$
The car will travel 600 km.

Key Concept: Ratio Comparison

c) 20 spoons
[Solution Description]
The given ratio is $8 : 12$ for sugar to glasses. To find the sugar needed for 30 glasses, set up the proportion:
$\frac{8}{12} = \frac{x}{30}$. Cross-multiplying gives $12x = 240$, so $x = 20$. Thus, 20 spoons of sugar are needed.

Key Concept: Ratio Comparison in Real-life Application

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the price per mL decreases with increasing volume, which is true because larger sizes generally have lower per-unit costs due to reduced packaging and handling expenses. The reason correctly explains this by mentioning economies of scale, which refers to cost advantages gained through increased production efficiency. Thus, both are true, and the reason correctly explains the assertion.

Key Concept: Proportional Reasoning

c) 20 kg
[Solution Description]
The ratio of fertilizer to land is $8 : 2$, which simplifies to $4 : 1$. For 5 acres, the required fertilizer is $4 \times 5 = 20$ kg.

Key Concept: Ratio application, Cost calculation

d) \$880
[Solution Description]
Total parts = $2 + 7 = 9$.
Cement quantity = $(2/9) \times 180 = 40$ kg.
Sand quantity = $(7/9) \times 180 = 140$ kg.
Cement cost = $40 \times 15 = \$600$.
Sand cost = $140 \times 2 = \$280$.
Total cost = $600 + 280 = \$880$.

Key Concept: Proportional Reasoning Application

c) \$36
[Solution Description]
First, find the cost per pen by dividing the total cost by the number of pens: $\frac{\$15}{5} = \$3$ per pen. Then, multiply the cost per pen by the number of pens needed: $\$3 \times 12 = \$36$. Therefore, 12 pens cost \$36.

Key Concept: Trairasika – Rule of Three

d) 24
[Solution Description]
Using the Rule of Three, identify the proportional relationship and solve for the unknown.
Given ratio: $8 : 12 :: 16 : x$.
Step 1: Find the factor of change between the known terms:
$16 ÷ 8 = 2$
Step 2: Apply the same factor to the second term to find $x$:
$12 × 2 = 24$
Therefore, $x = 24$.

Key Concept: Identifying Proportional Relationships

a) Yes, because $\frac{4}{5} = \frac{16}{20}$
[Solution Description]
To determine if the ratios are proportional, simplify both ratios to their lowest terms. The ratio $4 : 5$ is already simplified. For the ratio $16 : 20$, divide both terms by their HCF, which is 4: $16 ÷ 4 = 4$ and $20 ÷ 4 = 5$. Thus, the simplified form of $16 : 20$ is $4 : 5$. Since both ratios are equal after simplification, they are proportional.

Key Concept: Identifying proportional relationships, Applying factor of change

c) 25 liters
[Solution Description]
First, find the petrol consumption per kilometer: $\frac{15 \text{ liters}}{180 \text{ km}} = 0.0833$ liters/km. Then, multiply this rate by the new distance: $0.0833 \text{ liters/km} \times 300 \text{ km} = 25$ liters.

Key Concept: Simplifying Ratios

b) $2 : 3$
[Solution Description]
To simplify the ratio $24 : 36$, we find the HCF of 24 and 36, which is 12. Divide both terms by 12:
$24 ÷ 12 = 2$
$36 ÷ 12 = 3$
The simplified ratio is $2 : 3$.

Key Concept: Proportional Ratios

d) 18
[Solution Description]
To solve for $x$, we set up the proportion $12 : 18 :: x : 27$.
Cross-multiplying gives: $12 \times 27 = 18 \times x$.
Simplifying, we get $324 = 18x$.
Dividing both sides by 18: $x = 18$.

Key Concept: Proportional Ratios

a) $15 : 21$
[Solution Description]
Two ratios are proportional if their simplest forms are equal. The ratio $5 : 7$ is already in its simplest form. Check each option by simplifying it:
Option a) $15 : 21$ simplifies to $5 : 7$ (HCF = 3).
Option b) $10 : 14$ simplifies to $5 : 7$ (HCF = 2).
Option c) $20 : 28$ simplifies to $5 : 7$ (HCF = 4).
Option d) $25 : 30$ simplifies to $5 : 6$ (HCF = 5).
Thus, options a), b), and c) are proportional to $5 : 7$, but only one is needed as the correct answer. Here, $15 : 21$ is chosen.

Key Concept: Ratio simplification, proportional reasoning

a) $9:12$
[Solution Description]
To find the proportional ratio, first simplify $45:60$ by dividing both terms by their HCF (15):
$\frac{45}{15} : \frac{60}{15} = 3 : 4$
Now check which option simplifies to $3:4$.

Key Concept: Trairasika — The Rule of Three

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that for proportional ratios $a : b :: c : d$, the product of the means ($b$ and $c$) equals the product of the extremes ($a$ and $d$), i.e., $ad = bc$. This is a fundamental property of proportional ratios. The reason explains that cross multiplication ($ad = bc$) is a method to verify proportionality between two ratios. Since the reason correctly explains why the assertion holds true, both are correct and logically connected.

Key Concept: Comparative proportional analysis

b) Factory A is more efficient
[Solution Description]
Factory A good units = $500 \times 0.75 = 375$ in 3 hours → 125/hour.
Factory B good units = $800 \times 0.70 = 560$ in 5 hours → 112/hour.
Factory A has higher efficiency (125 vs 112 good units/hour).

Key Concept: Trairasika — The Rule of Three

c) 18
[Solution Description]
The problem involves inverse proportion, so we set it as $6 \times 12 = x \times 4$. Solving for $x$ gives $x = \frac{72}{4} = 18$. Thus, 18 workers are needed to complete the task in 4 days.

Key Concept: Comparative Proportion

b) No, the second shop is cheaper
[Solution Description]
To check proportionality, compare the cost per notebook for both shops.
First shop: Cost per notebook = $\frac{\$120}{3} = \$40$ per notebook.
Second shop: Cost per notebook = $\frac{\$190}{5} = \$38$ per notebook.
Since $\$40 \neq \$38$, the two ratios are not proportional. The second shop offers notebooks at a cheaper rate (\$38 vs. \$40), so it is the better deal.

Key Concept: Cross Multiplication Method

b) 21
[Solution Description]
Given the proportion $5:7 :: 15:x$. Using cross multiplication, we get:
$5 \times x = 7 \times 15$
Solving for $x$:
$x = \frac{7 \times 15}{5} = \frac{105}{5} = 21$

Key Concept: Cross Multiplication Method

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the cross multiplication method is used to find the fourth proportional in a proportion, which is true. The reason given is that in the proportion $a : b :: c : d$, the product of the extremes ($a$ and $d$) equals the product of the means ($b$ and $c$). This is also true, as it defines the cross multiplication rule $ad = bc$. Moreover, the reason correctly explains why the assertion is valid since the cross multiplication method is derived from this property. Therefore, both the Assertion and Reason are true, and the Reason is the correct explanation of the Assertion.

Key Concept: Mixing ratios, Proportionality

b) 5 liters
[Solution Description]
Current mixture has $(5/8)\times40 = 25$ liters alcohol and $(3/8)\times40 = 15$ liters water. Let $x$ liters of water be added. New ratio is:
$\frac{25}{15 + x} = \frac{5}{4}$
Cross multiplying:
$25 \times 4 = 5 \times (15 + x)$
$100 = 75 + 5x \Rightarrow 5x = 25 \Rightarrow x = 5 \text{ liters}$

Key Concept: Time-Distance, Proportionality

c) 7.5 days
[Solution Description]
This is an inverse proportion problem since more workers will take less time. The product of workers and days remains constant. Let $x$ be the required number of days.
$5 \text{ workers} \times 12 \text{ days} = 8 \text{ workers} \times x \text{ days}$
$x = \frac{5 \times 12}{8} = \frac{60}{8} = 7.5 \text{ days}$

Key Concept: Proportional Ratios

b) 40
[Solution Description]
Model the problem as the proportion $5 : 25 :: 8 : x$.
Step 1: Write the proportion and use the Rule of Three:
$5 \times x = 25 \times 8$
Step 2: Multiply the known terms:
$5x = 200$
Step 3: Solve for $x$ by dividing both sides by 5:
$x = \frac{200}{5} = 40$
The machine can produce 40 toys in 8 hours.

Key Concept: Proportional Ratios and Cross-Multiplication Method

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve the assertion, apply the cross-multiplication rule to the given proportion $6 : 10 :: 18 : x$.
Step 1: Write the proportion as $\frac{6}{10} = \frac{18}{x}$.
Step 2: Cross-multiply to get $6x = 180$.
Step 3: Solve for $x$ by dividing both sides by 6: $x = \frac{180}{6} = 30$.
Thus, the Assertion is true. The Reason correctly explains the Assertion because it provides the mathematical basis (cross-multiplication rule) for solving such proportional problems. Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Cross Multiplication

c) 8
[Solution Description]
Using the cross-multiplication method for proportions:
Given proportion is $4 : 9 :: x : 18$.
This can be written as $\frac{4}{9} = \frac{x}{18}$.
Cross-multiplying gives $4 \times 18 = 9 \times x$.
Calculating:
$72 = 9x$.
Solving for $x$:
$x = 72 / 9 = 8$.
Thus, the value of $x$ is 8.

Key Concept: Rule of Three Proportion

b) 10 workers
[Solution Description]
We use the rule of three to solve this problem. Let the number of workers be $x$. The work remains constant, so we have an inverse proportion between workers and days.
Given:
5 workers $\times$ 20 days = $x$ workers $\times$ 10 days.
To find $x$, divide both sides by 10 days:
$(5 \times 20) / 10 = x$.
Simplifying:
$100 / 10 = 10$.
Therefore, 10 workers are needed.

Key Concept: Rule of Three, Inverse Proportion

b) 4 days
[Solution Description]
This is an inverse proportion problem. The product of workers and days remains constant: $8 \times 6 = 12 \times d$. Solving for $d$:
$d = \frac{8 \times 6}{12} = 4 \text{ days}$

Key Concept: Rule of Three

c) 4 hours
[Solution Description]
Since both workers have the same efficiency, working together means their rates add up. The first worker’s rate is $\frac{1}{8}$ tasks/hour. Combined rate is $\frac{1}{8} + \frac{1}{8} = \frac{2}{8} = \frac{1}{4}$ tasks/hour. Thus, time taken together is $\frac{1}{\frac{1}{4}} = 4$ hours.

Key Concept: Rule of Three, Mixed Numbers

c) 30 liters
[Solution Description]
Convert mixed numbers to improper fractions: Pramāṇa = $\frac{13}{4}$ kg, Phala = $\frac{15}{2}$ liters, Ichchhā = 13 kg. Using the Rule of Three:
$ichchhāphala = \frac{phala \times ichchhā}{pramāṇa} = \frac{\frac{15}{2} \times 13}{\frac{13}{4}} = \frac{195/2}{13/4} = \frac{195 \times 4}{2 \times 13} = \frac{780}{26} = 30 \text{ liters}$

Key Concept: Proportionality

b) \$300
[Solution Description]
Using the Rule of Three:
$pramāṇa = 15$, $phala = 225$, $ichchhā = 20$. So,
$ichchhāphala = \frac{phala \times ichchhā}{pramāṇa} = \frac{225 \times 20}{15} = 300$
Hence, 20 notebooks cost \$300.

Key Concept: Mixture Ratio

b) 70 kg
[Solution Description]
Total parts in the ratio = $7 + 1 = 8$. Size of each part = $\frac{80}{8} = 10$ kg. Sugar corresponds to 7 parts, so quantity of sugar = $7 \times 10 = 70$ kg.

Key Concept: Dividing a mixture in ratio

c) 28 kg
[Solution Description]
Total weight = 40 kg
Ratio = 7:3
Total parts = 7 + 3 = 10
Size of each part = $40 \div 10 = 4$ kg
Flour quantity = $7 \times 4 = 28$ kg.

Key Concept: Sharing in given ratio

b) 12 and 6
[Solution Description]
Total chocolates = 18
Ratio = 2:1
Total parts = 2 + 1 = 3
Size of each part = $18 \div 3 = 6$
First friend’s share = $2 \times 6 = 12$ chocolates
Second friend’s share = $1 \times 6 = 6$ chocolates.

Key Concept: Sharing profit based on investment ratio

b) \$2500 and \$1500
[Solution Description]
Investment ratio = 50000 : 30000 = 5:3
Total parts = 5 + 3 = 8
Size of each part = $4000 \div 8 = 500$
First partner’s profit = $5 \times 500 = \$2500$
Second partner’s profit = $3 \times 500 = \$1500$.

Key Concept: Assertion-Reason

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that to divide \Rs.5,000 in the ratio $3 : 2$, we use the formula $\frac{x}{m + n}$.
The Reason explains that the formula $\frac{x}{m + n}$ helps find the size of each part when dividing a quantity in a given ratio.
Here, both Assertion and Reason are true. However, while the formula $\frac{x}{m + n}$ is indeed used to find the size of one part, it alone does not fully explain how the division is done in the ratio $3 : 2$. The complete process involves multiplying this value by the respective ratio terms ($3$ and $2$). Hence, the Reason does not correctly explain the Assertion.

Key Concept: Dividing quantities in a given ratio

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To solve this, let’s analyze the assertion and reason step by step.
1. The assertion states that for a ratio $3 : 5$, the larger part is $\frac{5}{8}$ of the total quantity. This is correct because:
Total parts = $3 + 5 = 8$.
Larger part corresponds to $5$ parts, so its fraction is $\frac{5}{8}$.
2. The reason claims that for any ratio $m : n$ where $n > m$, the larger part is $\frac{n}{m + n}$. This is also correct mathematically. However, the reason does not correctly explain the assertion because it generalizes the concept but doesn’t directly justify why the specific case ($3 : 5$) holds true.
Therefore, both assertion and reason are true, but the reason is not the correct explanation of the assertion.

Key Concept: Dividing quantities in a given ratio

c) Rs.720 and \Rs.480
[Solution Description]
To divide \Rs.1,200 in the ratio $3 : 2$:
$\text{Total parts} = 3 + 2 = 5$
$\text{Each part} = \frac{1200}{5} = Rs.240$
$\text{First share} = 3 \times 240 = Rs.720$
$\text{Second share} = 2 \times 240 = Rs.480$
The two shares are \Rs.720 and \Rs.480.

Key Concept: Dividing quantities in a given ratio, adjusting ratios

b) 5 liters
[Solution Description]
First, determine the current amounts of milk and water in the mixture.
Total parts in the original ratio 7:5 = 7 + 5 = 12 parts.
Amount of milk = $\frac{7}{12} \times 60 = 35$ liters.
Amount of water = $\frac{5}{12} \times 60 = 25$ liters.
Let $x$ be the liters of water to be added to change the ratio to 7:6.
The new amount of water will be $25 + x$, while milk remains 35 liters.
According to the new ratio:
$\frac{35}{25 + x} = \frac{7}{6}$.
Cross-multiplying:
$35 \times 6 = 7 \times (25 + x)$.
$210 = 175 + 7x$.
$35 = 7x$.
$x = 5$.
Therefore, 5 liters of water must be added.

Key Concept: Ratio division, Profit sharing

b) \$9,000
[Solution Description]
Total ratio parts = $5 + 3 = 8$. Each part corresponds to $\frac{24000}{8} = 3000$. Bina’s share is $3 \times 3000 = 9000$.

Key Concept: Sharing in Ratio, Verification of Parts

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion:
– Total amount (\$1,800) is to be divided in the ratio 5:4.
– Sum of ratio terms = 5 + 4 = 9.
– First part = $5 \times \frac{1800}{9} = 5 \times 200 = \$1,000$.
– Second part = $4 \times \frac{1800}{9} = 4 \times 200 = \$800$.
– Both parts match the assertion.
The reason states the correct formula for dividing a quantity in a given ratio, which is used to derive the assertion. Hence, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Dividing a Quantity in Given Ratio

b) 24 and 36
[Solution Description]
To divide 60 marbles in the ratio 2:3:
Total parts = $2 + 3 = 5$.
Size of each part = $\frac{60}{5} = 12$ marbles.
First friend’s share = $2 \times 12 = 24$ marbles.
Second friend’s share = $3 \times 12 = 36$ marbles.

Key Concept: Sharing in a given ratio

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To divide Rs.500 in the ratio 2:3:
1. Total parts = 2 + 3 = 5
2. Value of each part = Rs.500 / 5 = Rs.100
3. Larger share = 3 × Rs.100 = Rs.300
The Assertion is true as the larger share is indeed Rs.300. The Reason is also true because the total parts are 5. However, the Reason does not explain why the larger share is Rs.300; it only states the total parts. Hence, Reason is NOT the correct explanation of Assertion.

Key Concept: Profit Sharing

c) Rs.3,000
[Solution Description]
The investment ratio of Prashanti to Bhuvan is $90000 : 30000 = 3 : 1$. Total parts = $3 + 1 = 4$. Each part = $\frac{12000}{4} = 3000$. Bhuvan’s share = $1 \times 3000 = Rs.3000$.

Key Concept: Profit Sharing and Ratio Application

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we calculate the shares of A and B using the given ratio 5:3. The total profit is \Rs.12,000. The total parts of the ratio are 5 + 3 = 8. Each part corresponds to \Rs.12,000 / 8 = \Rs.1,500. Partner A’s share is 5 × \Rs.1,500 = \Rs.7,500, which matches the Assertion. The Reason correctly explains the method used to calculate the shares, as it describes the formula where the total profit is multiplied by the individual ratio divided by the sum of ratio parts.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Mixture Problem

d) 20 kg
[Solution Description]
Given ratio = 4:1, total parts = 4 + 1 = 5.
Sugar quantity = $\frac{4}{5} \times 25 = 20$ kg.

Key Concept: Profit sharing, Ratio division

c) Rs.15,000 and Rs.9,000
[Solution Description]
The investment ratio is 5:3. Total parts = 5 + 3 = 8.
Each part of profit = $\frac{24000}{8} = Rs.3,000$.
Prashanti’s share = $5 \times 3000 = Rs.15,000$.
Bhuvan’s share = $3 \times 3000 = Rs.9,000$.

Key Concept: Temperature conversion, Multi-step reasoning

c) $19^\circ \text{F}$
[Solution Description]
First, convert $-10^\circ \text{C}$ to Fahrenheit using the formula: $\frac{9}{5} \times (-10) + 32 = -18 + 32 = 14^\circ \text{F}$.
Then add the instrument error of $+5^\circ \text{F}$: $14 + 5 = 19^\circ \text{F}$.
The final temperature reading will be $19^\circ \text{F}$.

Key Concept: Unit Conversion – Volume

b) 18.5 kg
[Solution Description]
The mass ratio of gold to water is $37 : 2$. Since 1 litre of water weighs 1 kg, we can find the mass of 1 litre of gold using the ratio:
$\text{Mass of gold} = \left( \frac{37}{2} \right) \times 1 \text{ kg} = 18.5 \text{ kg}$
Thus, the mass of 1 litre of gold is 18.5 kg.

Key Concept: Unit Conversions – Temperature

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Step 1: Given the assertion that $68^\circ F$ is equal to $20^\circ C$.
Step 2: Verify the assertion by applying the reason formula $Celsius = \frac{5}{9} \times (Fahrenheit – 32)$.
Step 3: Substitute $68$ for Fahrenheit in the equation:
$\frac{5}{9} \times (68 – 32) = \frac{5}{9} \times 36 = 20$.
Step 4: The calculation confirms that $68^\circ F$ indeed equals $20^\circ C$, so both Assertion and Reason are true.
Step 5: Moreover, the Reason correctly explains how to derive Celsius from Fahrenheit, making it the correct explanation of the Assertion.

Key Concept: Unit Conversion – Length

c) 32.81 feet
[Solution Description]
Using the conversion factor: $1 \text{ metre} = 3.281 \text{ feet}$. To convert 10 metres to feet:
$10 \text{ metres} \times 3.281 \text{ feet/metre} = 32.81 \text{ feet}$
Hence, 10 metres is equal to 32.81 feet.

Key Concept:

c) 2,500 mL
[Solution Description]
Given, $1 \text{ litre} = 1,000 \text{ mL}$. Therefore, $2.5 \text{ litres} = 2.5 \times 1,000 \text{ mL} = 2,500 \text{ mL}$.

Key Concept: Unit Conversions – Temperature

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
Assertion (A) states that $32^\circ F$ is equal to $0^\circ C$, which is true as per the given conversion in the syllabus. Reason (R) provides the correct formula to convert Celsius to Fahrenheit, which is also true. However, Reason (R) does not explain why $32^\circ F$ is equal to $0^\circ C$. Thus, both are true, but Reason is not the correct explanation of Assertion.

Key Concept:

b) 2.471 acres
[Solution Description]
Given, $1 \text{ hectare} = 2.471 \text{ acres}$. So, the plot is equal to 2.471 acres.

Key Concept: Length, Area

b) 16146 square feet
[Solution Description]
First, calculate the area in square meters:
$Area = Length \times Width = 50 m \times 30 m = 1500 \text{ m}^2$.
Now, convert square meters to square feet using the conversion factor:
$1 \text{ m}^2 = 10.764 \text{ ft}^2$
So, $1500 \text{ m}^2 \times 10.764 \text{ ft}^2/\text{m}^2 = 16146 \text{ ft}^2$.

Key Concept: Unit Conversion in Proportional Reasoning

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that ensuring consistency in units is essential for accurate proportional reasoning. This is true because comparing or calculating with different units leads to incorrect results. The reason explains why this is necessary—different units cause wrong ratios and conclusions. Thus, both statements are true, and the reason correctly explains the assertion.

Key Concept: Time and Volume Conversion

c) 300 seconds
[Solution Description]
First, convert 10 litres to millilitres: $10\,\text{L} = 10,000\,\text{mL}$. Using the given rate, the time taken is calculated as:
$\text{Time taken} = \frac{15 \times 10,000}{500} = 300\,\text{seconds}$

Key Concept: Unit Conversion – Area

b) 4.1322 acres
[Solution Description]
First, calculate the area of the plot in square feet:
$300 \text{ ft} \times 600 \text{ ft} = 180,000 \text{ sq ft}$
Next, convert square feet to acres using the conversion factor:
$\frac{180,000 \text{ sq ft}}{43,560 \text{ sq ft/acre}} \approx 4.1322 \text{ acres}$

Key Concept: Unit Conversions, Proportional Reasoning

c) 27.55 litres
[Solution Description]
First, convert the area of the field from square feet to acres. The area is $800 \times 600 = 480,000$ square feet. Since $1$ acre = $43,560$ square feet, the area in acres is $\frac{480,000}{43,560} \approx 11.02$ acres.
Next, use the given ratio: $5$ litres for $2$ acres. The amount of diesel needed for $11.02$ acres can be calculated using proportionality:
$\text{Diesel} = \left( \frac{5 \text{ litres}}{2 \text{ acres}} \right) \times 11.02 \text{ acres} \approx 27.55 \text{ litres}$
Therefore, approximately $27.55$ litres of diesel are needed.