We will learn four numbers in proportion.
In general, a, b, c, d are said to be in proportion if a : b = c : d which is written as a : b :: c : d and read as a is to b as c is to d.
Here, a, b, c, d are respectively called the first, second, third and fourth term.
The first term a and the fourth term d are called the extreme terms or extremes.
The second term b and the third term c are called the middle terms or means.
If a, b, c, d are in proportions, then product of extremes = product of means
⟹ a × d = b × c
For example, in proportion 2 : 3 :: 24 : 36
Product of extremes = 2 × 36 = 72
Product of means = 3 × 24 = 72
Hence, product of extremes = product of means.
If the two given ratios are not equal, then they are not in proportions and the product of their extremes is not equal to the product of their means.
1. Are 50, 75, 30, 45 in proportion?
Solution:
50 : 75 = \(\frac{50}{75}\) = \(\frac{50 ÷ 25}{75 ÷ 25}\) = \(\frac{2}{3}\) = 2 : 3
30 : 45 = \(\frac{30}{45}\) = \(\frac{30 ÷ 15}{45 ÷ 15}\) = \(\frac{2}{3}\) = 2 : 3
50 : 75 = 30 : 45
Hence, 50, 75, 30, 45 are in proportion.
Alternative Method:
Product of extremes = 50 × 45 = 2250
Product of means = 75 × 30 = 2250
Product of extremes = Product of means
Hence, 50, 75, 30, 45 are in proportion.
2. Are the ratios 10 minutes : 1 hour and 6 hours: 36 hours in proportion?
Solution:
10 minutes : 1 hour = 10 minutes : 60 minutes
= 10 : 60
= \(\frac{10}{60}\)
= \(\frac{10 ÷ 10}{60 ÷ 10}\)
= \(\frac{1}{6}\)
= 1 : 6
and 6 hours : 36 hours = 6 : 36
= \(\frac{6}{36}\)
= \(\frac{6 ÷ 6}{36 ÷ 6}\)
= \(\frac{1}{6}\)
= 1 : 6
Therefore, 10 minutes : 1 hour = 6 hours : 36 hours
Hence, the ratios 10 minutes : 1 hour and 6 hours : 36 hours are in proportion.
2. Christopher drives his car at a constant speed of 12 km per 10 minutes. How long will he take to cover 48 km?
Solution:
Let Christopher take x minutes to cover 48 km.
Speed (in km) |
Time (in minutes) | |
12 |
10 | |
48 |
x |
Now, 12 : 10 :: 48 : x
⟹ 12x = 10 × 48
⟹ x = \(\frac{10 × 48}{12}\)
⟹ x = \(\frac{480}{12}\)
⟹ x = 40
Therefore, x = 40 minutes.
3. Are the ratios 45 km: 60 km and 12 hours: 15 hours form a proportion?
Solution:
45 km : 60 km = \(\frac{\textrm{45 km}}{\textrm{60 km}}\) = \(\frac{45}{60}\) = \(\frac{45 ÷ 15}{60 ÷ 15}\) = 3 : 4
and 12 hours : 15 hours = \(\frac{\textrm{12 hours}}{\textrm{15 hours}}\) = \(\frac{12}{15}\) = \(\frac{12 ÷ 3}{15 ÷ 3}\) = 4 : 5
Since 3 : 4 ≠ 4 : 5, therefore the given ratios do not form a proportion.
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