FINDING THE MISSING VALUES IN DIRECT AND INVERSE PROPORTION

Find the missing variable:

Problem 1 :

y varies directly with x. If y = -4 when x = 2, find y when x = -6.

Solution:

If y varies directly as x.

Then y = kx

-4 = k(2)

k = -4/2

k = -2

When x = -6

y = kx

= -2(-6)

y = 12

Problem 2 :

y varies inversely with x. If y = 40 when x = 16, find x when y = -5.

Solution:

If y varies inversely as x.

Then y = k/x

40 = k/16

k = 40 × 16

k = 640

when y = -5

-5 = 640/x

-5x = 640

x = -128

Problem 3 :

y varies inversely with x. If y = 7 when x = -4, find y when x = 5.

Solution:

If y varies inversely as x.

Then y = k/x

7 = k/-4

k = 7 × (-4)

k = -28

when x = 5

y = -28/5

y = -5.6

Problem 4 :

y varies directly with x. If y = 15 when x = -18, find y when x = 1.6.

Solution:

If y varies directly as x.

Then y = kx

15 = k(-18)

k = 15/(-18)

k = -5/6

When x = 1.6

y = kx

= -5/6(1.6)

y = -1.33

Problem 5 :

y varies directly with x. If y = 75 when x = 25, find x when y = 25.

Solution:

If y varies directly as x.

Then y = kx

75 = k(25)

k = 75/25

k = 3

When y = 25

y = kx

25 = 3x

x = 25/3

x = 8.33

Problem 6 :

If the cost of 9 toys is $333, find the cost of 16 such toys?

Solution:

The cost of 9 toys = $333

Cost of 9 is $333, then cost of 16 toys will be higher than $333. It comes under direct proportion.

Number of toys        Cost

            9                  333

          16                   x

Doing cross multiplication

9x = 16 (333)

x = 16(333)/9

x = 592

So, cost of 16 toys is $592.

Problem 7 :

If 22.5 m of a uniform iron rod weight 85.5 kg, what will be the length of 22.8 kg of the same rod?

Solution:

Mass per unit length of given rod is

Length of rod(m)           Weight(kg)

22.5                        85.5

x                            22.8

Since the weight decreases, its length will also decrease. Doing cross multiplication, we get

22.5(22.8) = x (85.5)

x = 22.5(22.8)/85.5

x = 6

So, length of the rod is 6 m.

Problem 8 :

Complete the following table given that x varies inversely as y.

Solution:

A x and y are in inverse variation. k to be the constant of inverse variation.

x = k/y

When x = 9 and y = 27

k = xy

k = 9 × 27

k = 243

When y = 9

k = xy

243 = 9x

x = 27

When x = 81

k = xy

243 = 81y

y = 3

Hence, the complete table is

Problem 9 :

If 15 oranges cost $70, find the cost of 39 oranges.

Solution:

Cost of 15 oranges = $70

cost of 39 oranges will be higher than $70. So, it comes under the concept of direct variation.

Number of oranges           Cost 

       15                          70

      39                            x

15x = 70(39)

x = 70(39)/15

x = 182

So, cost of 39 oranges is $182.

Problem 10 :

A farmer has enough food to feed 20 animals in his cattle for 6 days. How long the food last if would there were 25 animals in his cattle?

Solution:

Number of animals get food         Number of days

              20                                             6

              25                                             x

When number of animals increases then number of days will reduce

20 ⋅ 6 = 25 ⋅ x

x = 20(6)/25

x = 4.8

Then 4 days.

Problem 11 :

Reema types 540 words during half an hour. How many words would she type in 6 minutes?

Solution:

Number of days       Minutes

    540                          30

      x                             6

Since the number of minutes is reduced, then number of word will also reduce. Hence it is direct variation.

540(6) = 30x

x = 540(6)/30

x = 108

So, Reema can type 108 words in 6 minutes.

Problem 12 :

If 52 men can do a piece of work in 35 days, in how many men will do it in 14 days?

Solution:

Number of persons          Number of days

         52                                  35

          x                                  14

Number of persons should increase, then only the number of days will decrease. So, it comes under inverse proportion.

52 × 35 = x × 14

x = 1820/14

x = 130 men

So, 130 men will complete the work in 14 days.

Problem 13 :

If 12 m of a uniform iron rod weights 42 kg. What will be the weight of 6 m of same rod?

Solution:

Length of rod (m)         weight (kg)

      12                     42

       6                       x

If the length of rod is reduced then weight of rod will also reduce. So, it comes under direct variation.

12x = 42 (6)

x = 42(6)/12

x = 21 kg

So, length of 6 m rod is 21 kg.