Vinay Pacha

(i) Given numbers are 1.5, 5.5, 2.7, 9.9.
For example, to say the numbers are in proportion or not. We have to prove a: b = c: d.
Now, we find 1.5: 5.5 = 2.7: 9.9.
1.5: 5.5 = \(\frac{1.5}{5.5}\) = \(\frac{1.5×10}{5.5×10}\) = \(\frac{15}{55}\) = \(\frac{3}{11}\)
2.7: 9.9 = \(\frac{2.7}{9.9}\) = \(\frac{2.7×10}{9.9×10}\) = \(\frac{27}{99}\) = \(\frac{3}{11}\)
Thus, \(\frac{1.5}{5.5}\) = \(\frac{2.7}{9.9}\).
Therefore, the numbers 1.5, 5.5, 2.7, 9.9 are in proportion.

(ii) Given numbers are 2.5, 4.5, 1\(\frac{1}{4}\), 2\(\frac{1}{4}\).
As you know earlier, the numbers to be in proportion when a: b = c: d.
Now, 2.5: 4.5 = \(\frac{2.5}{4.5}\) = \(\frac{2.5×10}{4.5×10}\) = \(\frac{25}{45}\)= \(\frac{5}{9}\).
1\(\frac{1}{4}\): 2\(\frac{1}{4}\) = \(\frac{5}{4}\): \(\frac{9}{4}\) = \(\frac{5}{4}\)× 4: \(\frac{9}{4}\)× 4 = 5: 9 = \(\frac{5}{9}\).
Hence, \(\frac{5}{9}\) = \(\frac{5}{9}\).
Therefore, the numbers 2.5, 4.5, 1\(\frac{1}{4}\), 2\(\frac{1}{4}\) are in proportion.

(i) Given 4: 3 = 8: x
⇒ \(\frac{4}{3}\) = \(\frac{6}{x}\)
⇒ 4x = 8×3
⇒ 4x = 24
⇒ x = \(\frac{24}{4}\)
⇒ x = 6.
Thus, the value of x is 6.

(ii) Given 2.5: x = 8.1: 1.4
⇒ \(\frac{2.5}{x}\) = \(\frac{8.1}{1.4}\)
⇒ \(\frac{25}{10x}\) = \(\frac{81}{14}\)
⇒ 25 × 14 = 81 × 10x
⇒ 350 = 810x
⇒ x = \(\frac{810}{350}\)
⇒ x = 2.3
Hence, the x value is 2.3.

(iii) Given 1.0 : 2 = x: 4
⇒ \(\frac{1.0}{2}\) = \(\frac{x}{4}\)
⇒ \(\frac{10}{20}\) = \(\frac{x}{4}\)
⇒ 10 × 4 = 20x
⇒ 20x = 40
⇒ x = \(\frac{40}{20}\)
⇒ x = 2.
The value of x is 2.

(i) Given numbers are 0.2, 2.4, 4.0.
If we have three numbers to find whether they are continued proportion or not, thus a:b = b:a.
Now, 0.2: 2.4 = 2.4: 4.0
⇒ \(\frac{0.2}{2.4}\) = \(\frac{2.4}{4.0}\)
⇒ \(\frac{2}{24}\) = \(\frac{24}{40}\)
⇒ \(\frac{1}{12}\) ≠ \(\frac{3}{5}\)
Hence, the given numbers are not in continued proportion because 0.2: 2.4 ≠ 2.4: 4.0.

(ii) Given numbers are 4, 6, 9.
As we know, a:b = b:a
Now, 4: 6 = 6:9
⇒ \(\frac{4}{6}\) = \(\frac{6}{9}\)
⇒ \(\frac{2}{3}\) = \(\frac{2}{3}\)
Thus, the given numbers 4, 6, and 9 are in continued proportion because 4: 6 = 6: 9.

Given the numbers 8, 4, and m are in continued proportion.
8: 4 = 4: m
⇒ \(\frac{8}{4}\) = \(\frac{4}{m}\)
⇒ 8m = 16
⇒ m = \(\frac{16}{8}\)
⇒ m = 2.
Hence, the value of m is 2.

Given proportion numbers are 6, 2, and 9.
Let the fourth proportional be p.
Now, according to given problem 6, 2, 9, and p are proportionality.
Thus,
\(\frac{6}{2}\) = \(\frac{9}{p}\)
⇒ 6p = 9 × 2
⇒ 6p = 18
⇒ p = \(\frac{18}{6}\)
⇒ p = 3.
Therefore, the fourth proportional number is 3.

(i) Given proportion numbers 4, 6.
Now, find the third proportion.
Let the third proportion be k.
Thus,
\(\frac{4}{6}\) = \(\frac{6}{k}\)
⇒ 4k = 6 × 6
⇒ 4k = 36
⇒ k = \(\frac{36}{4}\)
⇒ k = 9.
Therefore, the third proportion number is 9.

(ii) Given proportion numbers are 1\(\frac{2}{3}\), \(\frac{4}{5}\)
Let the third proportion be k.
Thus,
\(\frac{5}{3}\) × \(\frac{5}{4}\) = \(\frac{4}{5}\) × \(\frac{1}{k}\)
⇒ \(\frac{25}{12}\) = \(\frac{4}{5k}\)
⇒ 25 × 5k = 4 × 12
⇒ 125k = 48
⇒ k = \(\frac{48}{125}\)
Hence, the third proportion number is \(\frac{48}{125}\).

(iii) Given proportion numbers are 3.4, 7.2.
Let the third proportion be k.
Now,
\(\frac{3.4}{7.2}\) = \(\frac{7.2}{k}\)
⇒ \(\frac{34}{72}\) = \(\frac{72}{10k}\)
⇒ 34 × 10k = 72 × 72
⇒ 340k = 5184
⇒ k = \(\frac{5184}{340}\)
⇒ k = 15.2
Thus, the third proportion value is 15.2.

(i) Given numbers are 10, q, 20, 15
To find the proportion numbers, we use a:b = c:d i.e., \(\frac{a}{b}\) = \(\frac{c}{d}\).
Now,
\(\frac{10}{q}\) = \(\frac{20}{15}\)
⇒ 10 × 15= 20q
⇒ 20q = 150
⇒ q = \(\frac{150}{20}\)
⇒ q = 7.5
Thus, the value of q is 7.5.

(ii) Given numbers are 8, 16, 24, q.
Now,
\(\frac{8}{16}\) = \(\frac{24}{q}\)
⇒ \(\frac{1}{2}\) = \(\frac{24}{q}\)
⇒ q = 24 × 2
⇒ q = 48
Hence, the value of q is 48.

(iii) Given numbers are q, 35, 15, 40.
We use the proportion formula, a:b = c:d i.e., \(\frac{a}{b}\) = \(\frac{c}{d}\).
Now,
\(\frac{q}{35}\) = \(\frac{15}{40}\)
⇒ 40q = 15 × 35
⇒ 40q = 525
⇒ q = \(\frac{525}{40}\)
⇒ q = 13.12
Therefore, the value of q for given proportional numbers is 13.12.

(i) Given x- y, x³- x²y
Now, to find the mean proportion.
Let p be the mean proportional between x- y, x³- x²y.
So, x- y: p:: p: x³- x²y
Product of extremes = Product of means.
Now,
p² = (x- y)(x³- x²y)
⇒ p² = x²(x- y)(x- y)
⇒ p² = x²(x- y)²
⇒ p = √(x²(x- y)²)
⇒ p = x(x- y)
Thus, the mean proportion of x- y, x³- x²y is x(x- y).

(ii) Given x³y, xy³
Let m be the mean proportion of x³y, xy³.
So, x³y: m:: m: xy³.
Product of extremes = Product of means.
Product of extremes = x³y × xy³
Product of means = m × m = m²
m² = x³y × xy³
⇒ m² = (x³ × x) × (y³ × y)
⇒ m² = x⁴ × y⁴
⇒ m = √(x⁴ × y⁴)
⇒ m = √(x²y²)²
⇒ m = x²y²
Hence, the mean proportion of x³y, xy³ is x²y².

(i) Given 8 and 32
Let the mean proportion between 8 and 32 is a.
Now, 8: a:: a: 32
We know that, Product of extremes = Product of means.
Here, extremes are 8 and 32 and means are a and a.
So, 8 × 32 = a × a
⇒ a² = 256
⇒ a = √256
⇒ a = 16
Thus, the value of mean proportion ‘a’ is 16.

(ii) Given 0.04 and 0.56
Let the mean proportion be p.
Now, 0.04: p:: p: 0.56
We know that, Product of extremes = Product of means.
The extremes are 0.04 and 0.56 and the means are p and p.
So, 0.04 × 0.56 = p × p
⇒ p² = 0.02
⇒ p = √0.02
⇒ p = 0.14
Therefore, the value of the mean proportion p is 0.14.

(iii) Given 4 and 25
Let the mean proportion between 4 and 25 be x.
Thus, 4: x:: x: 25
We know that, Product of extremes = Product of means.
Here, the extremes are 4 and 25 and the means are x and x.
So, 4 ×25 = x × x
⇒ x² = 100
⇒ x = √100
⇒ x = 10
Hence, the mean proportion of 4 and 25 is 10.

Given b is the mean proportion between a and c, then we have b² = ac.
Now, we have to prove that a⁴+ a²b²+ b⁴ / b⁴+ b²c²+ c⁴ = \(\frac{a²}{c²}\) i.e., L.H.S = R.H.S.
LHS = a⁴+ a²b²+ b⁴/ b⁴+ b²c²+ c⁴
Let us substitute b² = ac in LHS.
LHS = a⁴+ a²(ac)+ (ac)²/(ac)²+ (ac)c²+ c⁴
⇒ LHS = \(\frac{a²( a²+ ac+ c² )}{c²( a²+ ac+ c² )}\)
⇒ LHS = \(\frac{a²}{c²}\) = RHS
⇒ LHS = RHS
Therefore, a⁴+ a²b²+ b⁴/ b⁴+ b²c²+ c⁴= \(\frac{a²}{c²}\).

(i) Given 4\(\frac{4}{5}\), 2\(\frac{1}{2}\)
Now, change the mixed fraction into proper fraction
4\(\frac{4}{5}\) = \(\frac{24}{5}\)
2\(\frac{1}{2}\) = \(\frac{5}{2}\)
Let m be the mean proportion of \(\frac{24}{5}\), \(\frac{5}{2}\).
Product of extremes = Prouct of means
Here, the extremes are \(\frac{24}{5}\) and \(\frac{5}{2}\), the means are m and m.
\(\frac{24}{5}\) × \(\frac{5}{2}\) = m × m
⇒ m² = \(\frac{24}{2}\)
⇒ m² = 12
⇒ m = √12
⇒ m = 3.46
Thus, the value of mean proportion is 3.46.

(ii) Given a²b, ab²
Let k be the mean proportion of a²b and ab².
So, a²b: k:: k: ab².
In mean proportion, Product of extremes = Product of means.
Now, a²b × ab² = k²
⇒ k² = a³b³
⇒ k² = (ab)¹ × (ab)²
⇒ k = √((ab)¹)²
⇒ k = ab
Therefore, ab is the mean proportion of a²b, ab².

Given ratios are p: q = 4:6 and r: s = 3: 12
Now, to find the ratio of p: s.
As we know, if there is a ratio of quantities a: b and c: d. To get a: d = \(\frac{a}{b}\) × \(\frac{c}{d}\) = \(\frac{a}{d}\).
Similarly, we apply the same formula
p: s = \(\frac{p}{q}\) × \(\frac{r}{s}\)
⇒ p: s = \(\frac{4}{6}\) × \(\frac{3}{12}\)
⇒ p: s = 1: 6
Therefore, the value of the ratio p: s is 1: 6.

(i) Given x: y = 5: 4 and y: z = 2: 6
To get x: z, we use the formula, x: z = \(\frac{x}{y}\) × \(\frac{y}{z}\)
So,
x: z = \(\frac{5}{4}\) × \(\frac{2}{6}\)
⇒ x: z = \(\frac{5×2}{4×6}\)
⇒ x: z = \(\frac{10}{24}\)
⇒ x: z = \(\frac{5}{12}\)
Thus, the ratio x: z = 5: 12.

(ii) Given ratio x: y = 5: 4 and y:  = 2: 6
Now, to find the ratio of x: y: z.
Here, the values of y are not the same. To get the ratio of x: y: z, we have to multiply the value of y with a number to get both the values equal.
\(\frac{x}{y}\) = \(\frac{5}{4}\) × \(\frac{1}{1}\) = \(\frac{5}{4}\)
\(\frac{y}{z}\) = \(\frac{2}{6}\) × \(\frac{2}{2}\) = \(\frac{4}{12}\)
Thus, the ratio x: y: z = 5: 4: 12.

Given ratio \(\frac{4}{5}\): 1\(\frac{3}{5}\) i.e., \(\frac{4}{5}\): \(\frac{8}{5}\).
Therefore, the ratio becomes 4: 8 = 4+8 = 12.
Let the common multiple be x.
12x = 3876
x = \(\frac{3876}{12}\)
⇒ x = 323
Now, 4x = 4×323 = 1292
8x = 8×323 = 2584
Thus, the ratio is divided into ₹1292 and ₹2584.

(i) Given ratio a: b = 3: 7
The given equation is (2a+ 5b): (3a+ 6b).
Now, substitute the values of a and b in the above equation.
(2a+ 5b): (3a+ 6b) = ((2×3)+(5×7)): ((3×3)+(6×7)) = (6+35): (9+42) = 41: 51
Thus, the ratio of the equation (2a+ 5b): (3a+ 6b) is 41: 51.

(ii) Given equation is (a+ 2b): (7a- b)
The values of a and b were given are 3 and 7.
Now, substitute the values in the above equation.
(a+ 2b): (7a- b) = (3+(2×7)): ((7×3)-7) = (3+14): (21-7) = 15: 14
Hence, the ratio is 15: 14.

The reciprocal ratio of 12: 8 is 8: 12.
Now, 8: 12 = (m-1): (m+3)
⇒ \(\frac{8}{12}\) = \(\frac{m-1}{m+3}\)
⇒ 8(m+3) = 12(m-1)
⇒ 8m+24 = 12m-12
⇒ 8m- 12m = -12 – 24
⇒ -4m = -36
⇒ m = \(\frac{36}{4}\)
⇒ m = 9.
Therefore, the value of m is 9.

Given equation is 4x = 7y = 2z.
Now, 4x = 7y
⇒ x = \(\frac{7y}{4}\) —- (i)
7y = 2z
⇒ z = \(\frac{7y}{2}\) —- (ii)
Now,
x: y: z = \(\frac{7y}{4}\): y : \(\frac{7y}{2}\) (from (i) and (ii))
⇒ x: y: z = \(\frac{7}{4}\): 1 : \(\frac{7}{2}\)
⇒ x: y: z = 7: 4: 14
Hence, the ratio of x: y: z is 7: 4: 14.

Given ratio is \(\frac{1}{5}\): 1\(\frac{2}{3}\): \(\frac{2}{15}\) i.e., \(\frac{1}{5}\): \(\frac{5}{3}\): \(\frac{2}{15}\).
Let the common multiple be k.
Therefore, \(\frac{1k}{5}\): \(\frac{5k}{3}\): \(\frac{2k}{15}\) = 3570
⇒ \(\frac{3k}{15}\): \(\frac{25k}{15}\): \(\frac{2k}{15}\) = 3570
⇒ \(\frac{30k}{15}\) = 3570
⇒ 2k = 3570
⇒ k = \(\frac{3570}{2}\)
⇒ k = 1785
Now, to find the three equal ratios
\(\frac{1k}{5}\) = \(\frac{1×1785}{5}\) = \(\frac{1785}{5}\) =357
\(\frac{5k}{3}\) = \(\frac{5×1785}{3}\) = \(\frac{8925}{3}\) = 2975
\(\frac{2k}{15}\) = \(\frac{2×1785}{15}\) = \(\frac{3570}{15}\) = 238
The ratio is divided into $357, $2975, and $238.

Given ratios are a: b = 4: 1, b: c = 12: 7, and c: d = 7: 9
Now, to find the ratio of a: d.
Here,
a: d = \(\frac{a}{b}\) × \(\frac{b}{c}\) × \(\frac{c}{d}\)
⇒ a: d = \(\frac{4}{1}\) × \(\frac{12}{7}\) × \(\frac{7}{9}\)
⇒ a: d = \(\frac{4×12×7}{1×7×9}\)
⇒ a: d = \(\frac{4×4×1}{1×1×}\)
⇒ a: d = 16: 3
The ratio of a: d is 16: 3.
Triplicate ratio = 16³ : 3³ = 4096 : 27.
Therefore, the triplicate ratio of a: d = 4096: 27.

Given ratio is 13: 33
Now, we have to add the number so that we can get the ratio of 4: 9.
If we add the number 3 to the ratio of 13: 33
We got, 13+3: 33+3 = 16: 36.
So, the ratio becomes 16: 36 = 4: 9.
Thus, the number 4 should be added to the given ratio.

The given ratio is 4: 9.
Let the numbers be 4k and 9k
Given larger part of the number is 270.
Now, we find the number by solving 9k = 270.
k = \(\frac{270}{9}\)
⇒ k = 30
Another number is 4k, substitute the k value in 4k.
4k = 4 × 30 = 120.
Thus, the smaller part number is 120 and the actual number is 390.

Let the number of each denomination be 5x, 4x, and 2x respectively.
The amount of 3 dollars = 5x × 300 cents = 1500x cents
The amount of 50 cents = 4x × 50 cents = 200x cents
The amount of 4 dollars = 2x × 400 cents = 800x cents
The total amount given = 2450 × 100 cents =245000 cents.
Now, 1500x + 200x + 800x = 245000
⇒  2500x = 245000
⇒ x = \(\frac{245000}{2500}\)
⇒ x = 98
Now, we substitute the x value in each denomination.
The number of 3 dollars i.e., 5x = 5×98 = 490
The number of 50cents i.e., 4x = 4×98 = 392
The number of 4 dollars i.e., 2x = 2×98 = 196
Therefore, the denominations of each number are 490, 392, and 196.

Given two numbers in the ratio is 4: 6.
Let the two numbers be 4a and 6a.
Now, 4a+ 6a = 250
⇒ 10a = 250
⇒ a = \(\frac{250}{10}\)
⇒ a = 25
Next, we multiply the value of ‘a’ with the two numbers.
4a = 4×25 = 100
6a = 6×25 = 150
Thus, the numbers are 100 and 150.

The given number is 420 and the ratio is 3: 6: 5.
Now, to calculate the sum of the ratios = 3+6+5 = 14
Next, divide the three parts
First part = \(\frac{3}{14}\)×420 = 3×30 = 90
Second part = \(\frac{6}{14}\)×420 = 6×30 = 180
Third part = \(\frac{5}{14}\)×420 = 5×30 = 150
Hence, 420 can be divided into 90, 180, and 150 in the ratio of 3: 6: 5.

Let the hours of the workers worked to be in the ratio of 7: 6: 5: 6.
The sum of the ratio = 7+6+5+6 = 24hours
Now, get the amount per hour = Total wage / Hours worked = ₹14640/24 = ₹610 per hour
The wage of the first worker = 7×₹610 = ₹4270
The wage of the second worker = 6×₹610 = ₹3660
The wage of the third worker = 5×₹610 = ₹3050
The wage of the fourth worker = 6×₹610 = ₹3660
Therefore, each worker got the amount of ₹4270, ₹3660, ₹3050, and ₹3660.

Given quantity 4m 25cm
Now, first, we have to convert the given quantity into one unit.
As we know, 1m = 100cm
4m = 4×100 = 400cm
4m 25cm = 400cm + 25cm = 425cm.
Given ratio is 4: 6
The total number of parts = 4x+6x = 10x
Now, 10x = 425
⇒ x = \(\frac{425}{10}\) = 42.5
So,
The quantity of first number 4x = 4×42.5 = 170cm
The quantity of second number 6x = 6×42.5 = 255cm
Hence, the two quantities of the two numbers are 170cm and 255cm.

Given the ratio of length and the breadth is 8: 5 and the breadth is 85cm
Let the length be 8k and breadth be 5k.
Now, breadth 5k = 85cm
⇒ k = \(\frac{85}{5}\) = 17cm
Here, length = 8k
⇒ 8k = 8×17 = 136cm
Thus, the length of a rectangle is 136cm.

Given three parts ratio is 4: 3: 2.
Let the amount of money be 4x, 3x, and 2x.
From the given the first part is ₹448.
4x = ₹448
⇒ x = \(\frac{448}{4}\) = 112
Thus, x = 112
Now,
3x = 3×112 = 336 and 2x = 2×112 = 224
Therefore, the second amount = ₹336
the third amount = ₹224
The total amount of money = First amount + Second amount + Third amount = ₹448 +  ₹336 + ₹224 = ₹1008.
Hence, the total amount of money is ₹1008, the second part is ₹336, and the third part is ₹224.