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)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 answer
1 answer