QUE (1) From the following pairs of numbers, find the reduced form of ratio of first number to
second number. (i) 72, 60 (ii) 38,57 (iii) 52,78(i) 72, 60:
GCD of 72 and 60 is 12.
Dividing both numbers by 12:
72 / 12 = 6
60 / 12 = 5
So, the reduced form of the ratio is 6:5.
(ii) 38, 57:
GCD of 38 and 57 is 19.
Dividing both numbers by 19:
38 / 19 = 2
57 / 19 = 3
So, the reduced form of the ratio is 2:3.
(iii) 52, 78:
GCD of 52 and 78 is 26.
Dividing both numbers by 26:
52 / 26 = 2
78 / 26 = 3
So, the reduced form of the ratio is 2:3.
QUE (2) Find the reduced form of the ratio of the first quantity to second quantity.
(i) 700 `, 308 `
(ii) 14 `, 12 `. 40 paise.
(iii) 5 litre, 2500 ml
(iv) 3 years 4 months, 5 years 8 months
(v) 3.8 kg, 1900 gm (
vi) 7 minutes 20 seconds, 5 minutes 6 seconds.
Answer :
To find the reduced form of the ratio of various quantities, we need to ensure that the units are consistent for the quantities being compared. Let's solve each of these cases:
(i) 700 cm, 308 cm: The units are the same, so we can directly simplify the ratio: Ratio = 700 / 308 = 25 / 11 (This cannot be further reduced as 25 and 11 are coprime).
(ii) 14 cm, 12.40 paise: Since one quantity is in centimeters and the other is in paise (a unit of currency), we need to convert both quantities to a common unit. Let's convert everything to paise for consistency: 14 cm = 14 * 40 = 560 paise Ratio = 560 / 12.40 = 140 / 3.1 To simplify the ratio, divide both numbers by their greatest common divisor, which is 10: Ratio = 14 / 0.31 = 140 / 31 (This cannot be further reduced).
(iii) 5 liters, 2500 ml: Since the units are different (liters and milliliters), we need to convert them to a common unit. Let's convert liters to milliliters: 5 liters = 5 * 1000 ml = 5000 ml Ratio = 5000 / 2500 = 2 (This is already in reduced form).
(iv) 3 years 4 months, 5 years 8 months: First, convert everything to months for consistency: 3 years 4 months = 3 * 12 + 4 = 40 months 5 years 8 months = 5 * 12 + 8 = 68 months Ratio = 40 / 68 = 10 / 17 (This cannot be further reduced).
(v) 3.8 kg, 1900 gm: Since the units are different (kilograms and grams), we need to convert them to a common unit. Let's convert kilograms to grams: 3.8 kg = 3.8 * 1000 gm = 3800 gm Ratio = 3800 / 1900 = 2 (This is already in reduced form).
(vi) 7 minutes 20 seconds, 5 minutes 6 seconds: First, convert everything to seconds for consistency: 7 minutes 20 seconds = 7 * 60 + 20 = 440 seconds 5 minutes 6 seconds = 5 * 60 + 6 = 306 seconds Ratio = 440 / 306 = 220 / 153 (This cannot be further reduced).
So, the reduced forms of the given ratios are: (i) 25:11 (ii) 140:31 (iii) 2 (iv) 10:17 (v) 2 (vi) 220:153
QUE (3) Express the following percentages as ratios in the reduced form.
(i) 75 : 100 (ii) 44 : 100 (iii) 6.25% (iv) 52 : 100 (v) 0.64%To express the given percentages as ratios in reduced form, we need to convert the percentages to fractions and then simplify the fractions. Here's how you can do it:
(i) 75%: This can be written as a fraction 75/100, which simplifies to 3/4. (ii) 44%: This can be written as a fraction 44/100, which simplifies to 11/25. (iii) 6.25%: This can be written as a fraction 6.25/100, but to simplify it, we first convert the decimal to a fraction: 6.25/100 = 625/10000. Then, simplify this fraction if possible.
GCD of 625 and 10000 is 125. Dividing both numbers by 125: 625 / 125 = 5 10000 / 125 = 80
So, the simplified fraction is 5/80.
(iv) 52%: This can be written as a fraction 52/100, which simplifies to 13/25. (v) 0.64%: This can be written as a fraction 0.64/100, but it's easier to simplify by removing the decimal point: 64/10000. Then, simplify the fraction if possible.
GCD of 64 and 10000 is 16. Dividing both numbers by 16: 64 / 16 = 4 10000 / 16 = 625
So, the simplified fraction is 4/625.
To summarize, the given percentages expressed as ratios in reduced form are: (i) 3:4 (ii) 11:25 (iii) 5:80 (iv) 13:25 (v) 4:625
Let's assume that the amount of work required to build the small house is represented by a "work unit."
Given that three persons can complete the work in 8 days, we can write the equation:
3 persons * 8 days = 1 work unit
Now, we want to find out how many persons are required to complete the same work in 6 days. Let's denote the number of persons required as "x."
So, we have:
x persons * 6 days = 1 work unit
Now, since the amount of work is the same in both cases (building the same house), we can set the two equations equal to each other:
3 * 8 = x * 6
Solving for x:
x = (3 * 8) / 6 x = 24 / 6 x = 4
Therefore, to build the same house in 6 days, 4 persons are required.
To convert ratios to percentages, you can follow these steps:
- Divide the first number of the ratio by the second number.
- Multiply the result by 100 to get the percentage.
Let's calculate the percentages for each given ratio:
(i) 15 : 25 Percentage = (15 / 25) * 100 = 0.6 * 100 = 60%
(ii) 47 : 50 Percentage = (47 / 50) * 100 = 0.94 * 100 = 94%
(iii) 7 : 10 Percentage = (7 / 10) * 100 = 0.7 * 100 = 70%
(iv) 546 : 600 Percentage = (546 / 600) * 100 = 0.91 * 100 = 91%
(v) 7 : 16 Percentage = (7 / 16) * 100 = 0.4375 * 100 = 43.75%
So, the ratios as percentages are: (i) 60% (ii) 94% (iii) 70% (iv) 91% (v) 43.75%
Let's denote Abha's current age as "A" and her mother's current age as "M".
Given that the ratio of Abha's age to her mother's age is 2 : 5, we can write the equation:
A / M = 2 / 5
We are also given that at the time of Abha's birth, her mother's age was 27 years. This means that the age difference between Abha and her mother will always be 27 years.
So, when Abha was born, her mother's age was M - A = 27.
Now, we have a system of two equations:
- A / M = 2 / 5
- M - A = 27
Let's solve this system of equations. First, we can solve equation (2) for M: M = A + 27
Substitute this value of M into equation (1): A / (A + 27) = 2 / 5
Cross-multiply: 5A = 2(A + 27)
Distribute and solve for A: 5A = 2A + 54 3A = 54 A = 18
Now that we have Abha's age, we can find her mother's age using equation (2): M = A + 27 M = 18 + 27 M = 45
So, Abha's present age is 18 years, and her mother's present age is 45 years.
Let's denote the number of years after which the ratio of Vatsala's age to Sara's age becomes 5 : 4 as "x".
Currently, Vatsala's age is 14 years, and Sara's age is 10 years.
After x years, Vatsala's age will be 14 + x years, and Sara's age will be 10 + x years.
According to the given information, the ratio of their ages after x years will be 5 : 4:
(14 + x) / (10 + x) = 5 / 4
Cross-multiply to solve for x:
4(14 + x) = 5(10 + x)
Distribute on both sides:
56 + 4x = 50 + 5x
Subtract 4x from both sides:
56 = 50 + x
Subtract 50 from both sides:
x = 6
So, after 6 years, the ratio of Vatsala's age to Sara's age will become 5 : 4.