1)Find the value of y in the equation;243×3^(2y)÷729×3^(y)÷{3^(2y-1)
2)When a shirt is sold at sh 126 a loss of x% is made.If the same shirt is sold at sh154,a profit ofx %is realized.Find the buying price of the shirt.
3)Towns A and B are 429 km apart.Two lorries departed from A at the same time travelling.Lorry X travelled at an average speed of 15 km\h less than Y and reached 1 hour and 24 minutes later.
a) Calculate the average speed at lorry Y
b)How far was X from A when Y reached B.
c)A van left B heading towards A at the time lorries X and Y left A.If the van travelled at an average speed of 90 km\h ,how far from A did it meet lorry Y.
4)Mwangi and Otieno live 60 km apart.Mwangi leaves his home at 7 am, cycling towards Otienos house at 20 km\h.Otieno leaves home at 8 am, cycling towards Mwangis house at 8 km\h;
a)At what time did they meet?
b)How far is the meeting point from Mwangis house?
1) First simplify the denominator: 729*3^y = (9*9*9)*3^y = 3^(3y+2)
So the equation becomes:
243*3^(2y) / 3^(3y+2) * 1 / 3^(2y-1) =
243/3 * 1/3^(y+2) * 3^(1-2y) =
81/3^(y+1) * 3^(1-2y) =
3^(2-3y)
Therefore, y = (2- log3(81))/3 = 1/3
2) Let's call the buying price of the shirt B. We know that selling at sh 126 incurs a loss of x%, so the selling price is 0.01x*B + B = 1.01*B. Now we also know that selling at sh 154 results in a profit of x%, so the selling price is 1.01*B - 0.01x*(1.01*B) = (1 + 0.01x)*1.01*B.
We can equate the two selling prices: 1.01*B = (1 + 0.01x)*1.01*B
Simplifying the equation, we get: 1 = 1 + 0.01x
Therefore, x = 0, which means there is no profit or loss. So the buying price of the shirt B is equal to its selling price at sh 126, which is sh 126.
3) a) Let's call the average speed of lorry Y V. We know that lorry X travelled at V-15 km/h, and that it took 1 hour and 24 minutes longer to arrive at B. This is equivalent to 1 + 24/60 = 1.4 hours longer. Using the formula distance = speed × time, we have:
429 = V * t
429 = (V-15) * (t+1.4)
where t is the time it took for both lorries to meet. We can solve this system of equations to find V:
429 = V * t
429 = Vt - 15t + V*1.4 - 15*1.4
Substituting the first equation into the second one, we get:
429 = 429 - 15t + V * 1.4 - 15*1.4
Simplifying, we get: V = 75 km/h
b) Lorry Y traveled at an average speed of 75 km/h, so it took them 429/75 = 5.72 hours to reach B. In that time, lorry X travelled (V-15) * (t+1.4) = 60.36 km less than Y. Therefore, X was 429 - 60.36 = 368.64 km from A when Y reached B.
c) The van also traveled for 5.72 hours before it met lorry Y, so it covered a distance of 5.72 * 90 = 513.6 km from B. At that point, Y had also covered 513.6 km from A. Therefore, the distance between A and the meeting point is 429 - 513.6 = 84.6 km.
4) Let's call the distance between Mwangi's house and the meeting point d. We know that the total distance between them is 60 km, so the distance between the meeting point and Otieno's house is 60 - d. We can now set up two equations using the distance/speed/time formula:
d/20 = t (1)
(60-d)/8 = t-1 (2)
where t is the time it took for them to meet.
Simplifying equation (2), we get:
60 - d = 8t - 8
d = 8t - 52
Substituting this into equation (1), we get:
(8t-52)/20 = t
Simplifying, we get: t = 4 hours.
a) Mwangi left at 7 am, so he cycled for 4 hours until the meeting point at 11 am. Otieno left at 8 am, so he cycled for 3 hours until the meeting point at 11 am.
b) Substituting t = 4 into the equation for d, we get: d = 8*4 - 52 = 12 km. Therefore, the meeting point was 12 km from Mwangi's house.
1) To find the value of y in the equation 243×3^(2y)÷729×3^(y)÷{3^(2y-1), we can simplify step by step:
Step 1: Simplify 243×3^(2y) to get 729×3^(y).
Step 2: Simplify 729×3^(y) to get 3^(2y-1).
Step 3: Simplify {3^(2y-1) to get 3^(2y).
Step 4: Equate the two sides of the equation: 3^(2y) = 3^(2y).
Step 5: Since the bases are the same (3) and the powers are equal, we can equate the exponents: 2y = 2y.
Step 6: Subtract 2y from both sides of the equation: 0 = 0.
Therefore, the value of y can be any real number since the equation is always true.
2) Let's find the buying price of the shirt step by step:
Let the cost price (buying price) be 'C' and the loss/profit percentage be 'x'.
According to the given information:
Selling price at a loss of x% = sh 126.
Selling price at a profit of x% = sh 154.
Step 1: Calculate the selling price at a loss of x%:
sh 126 = (100 - x)% of C
126 = (100 - x)/100 * C
Step 2: Calculate the selling price at a profit of x%:
sh 154 = (100 + x)% of C
154 = (100 + x)/100 * C
Step 3: Divide the second equation by the first equation to eliminate C:
154/126 = [(100 + x)/100 * C] / [(100 - x)/100 * C]
Step 4: Simplify the equation:
154/126 = (100 + x)/(100 - x)
Step 5: Cross multiply:
154 * (100 - x) = 126 * (100 + x)
Step 6: Expand and simplify:
15400 - 154x = 12600 + 126x
Step 7: Rearrange the equation:
280x = 2800
Step 8: Divide by 280:
x = 10
Therefore, the buying price of the shirt is sh 126.
3) Let's solve the problem step by step:
a) To find the average speed of lorry Y, we can use the formula: distance = speed * time.
Given:
- Distance between towns A and B: 429 km.
- Lorry X travels at an average speed of 15 km/h less than Y.
- Lorry X reached 1 hour and 24 minutes later than lorry Y.
Let's denote the average speed of lorry Y as 'v' km/h.
Distance covered by lorry X = Distance covered by lorry Y + 429 km
(speed of lorry X) * (time taken by lorry Y) = v * (time taken by lorry Y + 1 hour and 24 minutes)
Since speed = distance / time, we have:
v * (time taken by lorry Y) = (v - 15) * (time taken by lorry Y + 1.4) [1.4 hours = 1 hour 24 minutes]
Simplifying the equation:
v * t = (v - 15) * (t + 1.4)
Solving this equation will give us the value of 'v'.
b) To find the distance of X from A when Y reached B, we can use the formula: Distance = speed * time.
The time taken by lorry Y to reach B is the same as the time taken by lorry X to reach X (since they departed at the same time).
Let's denote this time as 't' hours.
Distance of X from A = Distance of Y from A + Distance of Y from B
(speed of X) * t = v * t + 429 km
Simplifying the equation will give us the distance of X from A.
c) To find how far from A the van met lorry Y, we need to find the distance traveled by lorry Y while the van was traveling.
The time taken by the van to meet lorry Y is the same as the time taken by lorry Y to meet the van (since they both started at the same time).
Using the formula: Distance = Speed * Time, we can find the distance traveled by lorry Y.
Distance traveled by lorry Y = (speed of lorry Y) * (time taken to meet)
Simplifying the equation will give us the distance of the van from A when it met lorry Y.
4) Let's solve the problem step by step:
a) To determine the time when Mwangi and Otieno met, we need to find the time taken by each person by dividing the distance by their respective speeds.
Distance covered by Mwangi = Distance covered by Otieno
Speed of Mwangi * Time taken by Mwangi = Speed of Otieno * Time taken by Otieno
Since Otieno started one hour after Mwangi, the time taken by Otieno will be one hour less than the time taken by Mwangi.
Solving this equation will give us the time when they met.
b) To find the distance of the meeting point from Mwangi's house, we can use either Mwangi's time or Otieno's time to find the distance covered by either person.
Using Mwangi's time for simplicity:
Distance covered by Mwangi = Speed of Mwangi * Time taken by Mwangi
This gives us the distance from Mwangi's house to the meeting point.