Definitions [5]
A ratio is the relationship between two quantities of the same kind with the same unit, obtained by dividing the first by the second.
Example:
The ratio between 15 kg and 20 kg
15 kg : 20 kg = `15/20` = `3/4` = 3:4.
Four non-zero quantities, a, b, c, and d, are said to be in proportion (or are proportional) if:
a : b = c : d.
The above equation is expressed as a : b :: c : d
This is read as “a is to b as c is to d.”
Direct Proportion: Two quantities x and y are said to be in direct proportion if they increase or decrease together in such a manner that the ratio of their corresponding values remains constant.
The unitary method is a process used to find the value of a single unit from the value of multiple units and then find the value of multiple units from the value of a single unit.
Inverse Proportion:
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Two quantities x and y are said to be in inverse proportion if Two quantities may change in such a manner that if one quantity increases, the other quantity decreases, and vice versa.
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For example,
1) As the number of workers increases, the time taken to finish the job decreases.
2) If we increase the speed, the time taken to cover a given distance decreases.
- To understand this, let us look into the following situation.
Zaheeda can go to her school in four different ways. She can walk, run, cycle, or go by car. As Zaheeda doubles her speed by running, time reduces to half. As she increases her speed to three times by cycling, time decreases to one third. Similarly, as she increases her speed to 15 times, time decreases to one-fifteenth. (Or, in other words, the ratio by which time decreases is inverse of the ratio by which the corresponding speed increases).
Two quantities x and y are said to be in inverse proportion if Two quantities may change in such a manner that if one quantity increases, the other quantity decreases, and vice versa.
For example,
1) As the number of workers increases, the time taken to finish the job decreases.
2) If we increase the speed, the time taken to cover a given distance decreases.
Zaheeda can go to her school in four different ways. She can walk, run, cycle, or go by car. As Zaheeda doubles her speed by running, time reduces to half. As she increases her speed to three times by cycling, time decreases to one third. Similarly, as she increases her speed to 15 times, time decreases to one-fifteenth. (Or, in other words, the ratio by which time decreases is inverse of the ratio by which the corresponding speed increases).
Relation for Inverse Proportion:
Two quantities x and y are said to be in inverse proportion if an increase in x causes a proportional decrease in y (and vice-versa) in such a manner that the product of their corresponding values remains constant. That is, if xy = k, then x and y are said to vary inversely. In this case if y1, y2 are the values of y corresponding to the values x1, x2 of x respectively then x1y1 = x2y2 or `x_1/x_2 = y_2/y_1`.
Theorems and Laws [9]
If x : a = y : b, prove that `(x^4 + a^4)/(x^3 + a^3) + (y^4 + b^4)/(y^3 + b^3) = ((x + y)^4 + (a + b)^4)/((x+ y)^3 + (a + b)^3`
`x/a = y/b` = k (say)
x = ak, y = bk
L.H.S. = `(x^4 + a^4)/(x^3 + a^3) + (y^4 + b^4)/(y^3 + b^3)`
= `(a^4k^4 + a^4)/(a^3k^3 + a^3) + (b^4k^4 + b^4)/(b^3k^3 + b^3)`
= `(a^4(k^4 + 1))/(a^3(k^3 + 1)) + (b^4(k^4 + 1))/(b^3(k^3 + 1)`
= `(a(k^4 + 1))/(k^3 + 1) + (b(k^4 + 1))/(k^3 + 1)`
= `(a(k^4 + 1) + b(k^4 + 1))/(k^3 + 1)`
= `((k^4 + 1)(a + b))/(k^3 + 1)`
R.H.S. = `((x + y)^4 + (a + b)^4)/((x+ y)^3 + (a + b)^3`
= `((ak + bk)^4 + (a + b)^4)/((ak + bk)^3 + (a + b)^3`
= `(k^4(a + b)^4 + (a - b)^4)/(k^3(a + b)^3(a + b)^3`
= `((a + b)^4(k^4 + 1))/((a + b)^3(k^3 + 1)`
= `((a + b)(k^4 + 1))/(k^3 + 1)`
= `((k^4 + 1)(a + b))/(k^3 + 1)`
∴ L.H.S. = R.H.S.
Hence proved
If `x/a = y/b = z/c`, prove that `(3x^3 - 5y^3 + 4z^3)/(3a^3 - 5b^3 + 4c^3) = ((3x - 5y + 4z)/(3a - 5b + 4c))^3`.
`x/a = y/b = z/c` = k(say)
x = ak, y = bk, z = ck
L.H.S. = `(3x^3 5y^3 + 4z^3)/(3a^3 5b^3 + 4c^3)`
= `(3a^3k^3 - 5b^3k^3 + 4c^3k^3)/(3a^3 - 5b^3 + 4ac^3)`
= `(k^3(3a^3 - 5b^3 + 4c^3))/(3a^3 - 5b^3 + 4c^3`
= k3
R.H.S. = `((3x - 5y + 4z)/(3a - 5b + 4c))^3`
= `((3ak - 5bk + 4ck)/(3a - 5b + ac))^3`
= `((k(3a - 5b + 4c))/(3a - 5b + 4c))^3`
= (k)3
= k3
∴ L.H.S. = R.H.S.
If `a/c = c/d = e/f` prove that: `(a^3 + c^3)^2/(b^3 + d^3)^2 = e^6/f^6`
`a/c = c/d = e/f` = k(say)
∴ a = bk, c = dk, e =fk
L.H.S.
= `(a^3 + c^3)^2/(b^3 + d^3)^2`
= `(b^3k^3 + d^3k^3)^2/(b^3 + d^3)^2`
= `[k^3(b^3 + d^3)]^2/(b^3 + a^3)^2`
= `(k^6(b^3 + d^3)^2)/(b^3 + d^3)^2`
= k6
R.H.S. = `e^6/f^6`
= `f^6k^6/f^6`
= k6
∴ L.H.S. = R.H.S.
If x, y, z are in continued proportion, prove that: `(x + y)^2/(y + z)^2 = x/z`.
x, y, z are in continued proportion
Let `x/y = y/z = k`
Then y = kz
x = yk
= kz × k
= k2z
Now L.H.S.
= `(x + y)^2/(y + z)^2`
= `(k^2 z + kz)^2/(kz + z)^2`
= `{kz(k + 1)}^2/{z(k + 1)}^2`
= `(k^2z^2(k + 1)^2)/(z^2(k + 1)^2)`
= k2
R.H.S. = `x/z`
= `(k^2z)/z`
= k2
∴ L.H.S. = R.H.S.
If a, b, c, d are in continued proportion, prove that: `((a - b)/c + (a - c)/b)^2 - ((d - b)/c + (d - c)/b)^2 = (a - d)^2 (1/c^2 - 1/b^2)`.
a, b, c, d are in continued proportion
∴ `a/b = b/c = c/d` = k(say)
∴ c = dk, b = dk2, a = bk = dk2. k = dk3
L.H.S.
= `((a - b)/c + (a - c)/b)^2 - ((d - b)/c + (d - c)/b)^2`
= `((dk^3 - dk^2)/(dk) + (dk^3 - dk)/(dk^2))^2 - ((d - dk^2)/(dk) + (d - dk)/(dk^2))^2`
= `((dk^2(k - 1))/(dk) + (dk(k^2 - 1))/(dk^2))^2 - ((d(1 - k^2))/(dk) + (d( 1 - k^2))/(dk^2))^2`
= `((k(k - 1) + (k^2 - 1))/k)^2 - ((1 - k^2)/k + (1 - k)/k^2)^2`
= `((k^2(k - 1) + (k^2 - 1))/k)^2 - ((k (1- k^2) + 1 - k)/k^2)^2`
= `((k^3 - 1)^2)/k^2 - (-k^3 + 1)^2/k^4`
= `(k^3 - 1)^2/k^2 - (1 - k^3)^2/k^4`
= `((k^3 - 1)/k^2)^2 ((1 - 1)/k^2)`
= `((k^3 - 1)^2(k^2 - 1))/k^4`
= `((k^3 - 1)^2(k^2 - 1))/k^4`
R.H.S.
= `(a - d)^2(1 / c^2 - 1/b^2)`
= `(dk^3 - d)^2(1 / (d^2k^2) - (1)/(d^2k^4))`
= `d^2(k^3 - 1)^2((k^2 - 1)/(d^2k^4))`
= `((k^3 - 1)^2(k^2 - 1))/k^4`
∴ L.H.S. = R.H.S.
If x, y and z are in continued proportion, Prove that:
`x/(y^2.z^2) + y/(z^2.x^2) + z/(x^2.y^2) = 1/x^3 + 1/y^3 + 1/z^3`
Given: x, y and z are in continued proportion.
∴ `x/y = y/z`
⇒ y2 = xz
To prove: `x/(y^2.z^2) + y/(z^2.x^2) + z/(x^2.y^2) = 1/x^3 + 1/y^3 + 1/z^3`
Proof: Solving L.H.S.:
`x/(y^2.z^2) + y/(z^2.x^2) + z/(x^2.y^2)`
⇒ `(x^3 + y^3 + z^3)/(x^2.y^2.z^2)`
⇒ `(x^3 + y^3 + z^3)/(x^2.xz.z^2)`
⇒ `(x^3 + y^3 + z^3)/(x^3.z^3)`
⇒ `x^3/(x^3.z^3) + y^3/(x^3.z^3) + z^3/(x^3.z^3)`
⇒ `1/z^3 + y^3/(xz)^3 + 1/x^3`
⇒ `1/z^3 + y^3/(y^2)^3 + 1/x^3`
⇒ `1/z^3 + y^3/y^6 + 1/x^3`
⇒ `1/z^3 + 1/y^3 + 1/x^3`
Since L.H.S. = R.H.S.
Hence proved.
If a, b, c, d are in continued proportion, prove that: `(a^3 + b^3 + c^3)/(b^3 + c^3 + d^3) = a/d`
a, b, c, d are in continued proportion
∴ `a/b = b/c = c/d` = k(say)
∴ c = dk, b = ck = dk2, a = bk = dk3
L.H.S.
= `(a^3 + b^3 + c^3)/(b^3 + c^3 + d^3)`
= `((dk^3)^3 + (dk^2)^3 + (dk)^3)/((dk^2)^3 + (dk)^3 + d^3)`
= `(d^3k^9 + d^3k^6 + d^3k^3)/(d^3k^6 + d^3k^3 + d^3)`
= `(d^3k^3(k^6 + k^3 + 1))/(d^3(k^6 + k^3 + 1)`
= k3
R.H.S.
= `a/d`
= `(dk^3)/d`
= k3
∴ L.H.S. = R.H.S.
If a, b, c, d are in continued proportion, prove that: (a + d)(b + c) – (a + c)(b + d) = (b – c)2
a, b, c, d are in continued proportion
∴ `a/b = b/c = c/d` = k(say)
∴ c = dk, b = ck = dk. k = dk2,
a = bk = dk2. k = dk3
L.H.S. = (a + d)(b + c) – (a + c)(b + d)
= (dk3 + d) (dk2 + dk) – (dk3 + dk) (dk2 + d)
= d(k3 + 1) dk(k + 1) – dk (k2 + 1) d(k2 + 1)
= d2k(k + 1) (k3 + 1) – d2k (k2 + 1) (k2 + 1)
= d2k[k4 + k3 + k + 1 – k4 - 2k2 - 1]
= d2k[k3 – 2k2 + k]
= d2k2[k2 – 2k + 1]
= d2k2(k – 1)2
R.H.S. = (b – c)2
= (dk2 – dk)2
= d2k2(k – 1)2
∴ L.H.S. = R.H.S.
Hence proved.
If a + c = mb and `1/b + 1/d = m/c`, prove that a, b, c and d are in proportion.
a + c = mb and `1/b + 1/d = m/c`
a + c = mb ...(1)
`1/b + 1/d = m/c` ...(2)
Step 1: Simplify the second condition
`1/b + 1/d = m/c`
Take LCM of b and d:
`(d + b)/(bd) = m/c`
c(d + b) = mbd
cd + cb = mbd ...(3)
Step 2: Use the first condition
a + c = mb
Multiply both sides by d:
d(a + c) = mbd
ad + cd = mbd ...(4)
Step 3: Compare equations (3) and (4)
cd + cb = mbd
ad + cd = mbd
ad + cd = cd + cb
Subtract cdcdcd from both sides:
ad = cb
Step 4: Convert to ratio form
ad = bc
Divide both sides by bd:
`a/b = c/d`
Thus,
a : b = c : d
Hence, a, b, c and d are proportional.
