6TH GRADE MATH HELP - NUMBER OF SOLVED EXAMPLES AND EXERCISES

Please study

Algebra Word Problems before 6th Grade Math Help

if you have not already done so.

There we discussed The Method of Solving Word
Problems.That knowledge is a prerequisite here.

Example 1 of 6th Grade Math Help

One fifth of a number of butterflies in a garden are on jasmines
and one third of them are on roses. Three times the difference of
the butterflies on jasmines and roses are on lilys.If the remaining
one is flying freely, find the number of butterflies in the garden.

solution to Example 1 :

Let x be the number of butterflies in the garden.
As per data,
Number of butterflies on jasmines = x⁄5.
Number of butterflies on roses = x⁄3
then
difference of the butterflies on jasmines and roses = x⁄3 - x⁄5
As per data
Number of butterflies on lilys = Three times the difference of
the butterflies on jasmines and roses
= 3(x⁄3 - x⁄5)
As per data,
Number of butterflies flying freely = 1.
∴ number of butterflies in the garden = x
= Number of butterflies on jasmines + Number of butterflies on roses
+ Number of butterflies on lilys + Number of butterflies flying freely
= x⁄5 + x⁄3 + 3 (x⁄3 - x⁄5) + 1.
⇒ x = x⁄5 + x⁄3 + 3 x x⁄3 - 3 x x⁄5 + 1.

This is the Linear Equation formed by converting
the given word statements to the symbolic language.

Now we have to solve this equation.
⇒ x = x⁄5 + x⁄3 + x - 3x⁄5 + 1
Cancelling x, which is present on both sides, we get
0 = x⁄5 + x⁄3 - 3x⁄5 + 1
L.C.M. of the denominators 3, 5 is 3 x 5 = 15.
Multiplying both sides of the equation with 15, we get
15 x 0 = 15 x x⁄5 + 15 x x⁄3 - 15 x 3x⁄5 + 15 x 1
⇒ 0 = 3x + 5x - 3 x 3x + 15.
⇒ 0 = 8x - 9x + 15 ⇒ 0 = -x + 15 ⇒ 0 + x = 15
⇒ x = 15.
Number of butterflies in the garden = x = 15. Ans.

Check:
Number of butterflies on jasmines = x⁄5 = 15⁄5 = 3.
Number of butterflies on roses = x⁄3 = 15⁄3 = 5.
Number of butterflies on lilys = 3(5 - 3) = 3(2) = 6.
Number of butterflies flying freely = 1.
Total butterflies = 3 + 5 + 6 + 1. = 15. Same as the Ans.(verified.)

Example 2 of 6th Grade Math Help

A man rowed upstream for 4 hours and returned to his starting point
in 2 hours. If he rowed at the rate of 5km/hr. in still water, what was
the rate of flow of the stream.


solution to Example 2 of 6th Grade Math Help :

Let xkm/hr. be the rate of flow of the stream.
Then the rowing rate of the man upstream
= rowing rate of the man in still water - the rate of flow of the stream.
= 5km/hr. - xkm/hr. = (5 - x)km/hr.
And the rowing rate of the man downstream
= rowing rate of the man in still water + the rate of flow of the stream.
= 5km/hr. + xkm/hr. = (5 + x)km/hr.
We know
distance = speed x time.
As per data, time for upstream rowing = 4 hrs.
And time for downstream rowing = 2 hrs.
∴ distance rowed during upstream rowing = (5 - x)km/hr. x 4 hrs.
= 4(5 - x)km.
and distance rowed during downstream rowing = (5 + x)km/hr. x 2 hrs.
= 2(5 + x)km.
But the two distances are equal.
∴ 4(5 - x)km. = 2(5 + x)km.

This is the Linear Equation formed by converting
the given word statements to the symbolic language.

Now we have to solve this equation.
⇒ 4 x 5 - 4x = 2 x 5 + 2x
⇒ 20 - 4x = 10 + 2x ⇒ -4x - 2x = 10 - 20 ⇒ -6x = -10
⇒ x = (-10)⁄(-6) = 5⁄3.
thus rate of flow of the stream = x = (5⁄3)km/hr. Ans.

Check:
upstream distance = 4(5 - x)km = 4(5 - 5⁄3) = (4⁄3)(5 x 3 - 5)
= (4⁄3)(15 - 5) = (4⁄3)(10) = (40⁄3) km.
downstream distance = 2(5 + x)km = 2(5 + 5⁄3) = (2⁄3)(5 x 3 + 5)
= (2⁄3)(15 + 5) = (2⁄3)(20) = (40⁄3) km.
Both distances are equal. (verified.)

Exercise on 6th Grade Math Help

Solve the following problems on 6th Grade Math Help.

1. In a stream the current flows at the rate of 4 km/hr.
   For a boat the time taken to cover a certain distance
   upstream is 5 times the time it takes to cover the same distance
   downstream. Find the speed of the boat in still water.
2. In a stream whose current flows at the speed of 3km/hr,
   a man rowed upstream and returned to his starting point
   in 6.25 hours. If the man rows at a speed of 5 km/hr. in still water,
   how long did he row upstream ?

Answers to Exercise on 6th Grade Math Help

The Answers to the Problems on 6th Grade Math Help
of the exercise given above, are given below.

1. 6 km/hr
2. 5 hrs.