Review Innovations CE Review for Nov 2022 – Algebra 6
Clock-related Problems: 10. The number of revolutions per minute of a ball governor
If the minute hand of the clock required to keep the balls h inches below the point of
will describe an arc x minute suspension varies inversely as the square root of h. If the
spaces long, then during the
x balls are 6 inches below for 40 revolutions per minute, at
same time span, the hour hand what speed will they be 1-1/2 inches below?
x/12
will also describe an arc x/12 RI Answer: 80 rpm
minute spaces long. In short, to 11. A proposal was lost in which 600 persons voted. The
identify the corresponding same persons having voted again on the same proposal,
movement of the hour hand, it was carried by twice as many as it was before lost by,
always divide the movement of and the new majority is to the former as 8:7. How many
the minute hand by 12. changed their minds? Answer: 150
1. What time between 2 and 3 o’clock will the angle
between the hands of the clock be bisected by the line
connecting the center of the clock and the 3 o’clock mark? PROBLEMS FOR PRACTICE
Answer: 2:18:27.6 Situation 1. What time after 3:00 o'clock will the minute-
2. At what time between three and four o’clock is the hand and the hour-hand of the clock be
minute hand the same distance from VIII as the hour 1. together for the first time? 3:16:21.8
hand is from XII? 3:23.08 2. perpendicular for the first time? 3:32:43.6
3. in straight line for the first time? 3:49:05.4
3. It is now past 4PM. Four minutes from now, the minute
hand of the clock will be directly opposite the position of Situation 2. How soon after 5:00 o'clock will the hands of
the hour hand 8 minutes ago. What is the time now? the clock form…
4:49:27 4. a 60-degree angle for the first time? 5:16.4
5. a 60-degree angle for the second time? 5:38.2
6. a 150-degree angles? 5:54.5
Mixture Problems:
4. How much water must be evaporated from a 15-liter 12% 7. What time after 8:00 o-clock, will the minute-hand and
dye solution to obtain a solution that is 20% dye? the hour-hand of the clock, be at angle of 37.4° for the
Assume that the total amount of dye is not affected by second time? Ans: 8:50.44
the process of evaporation. Ans: 6 liters 8. If it were two hours later, it would be half as long until
5. A given alloy contains 20% copper and 5% tin. How midnight as it would be if it were an hour later. What
many pounds of copper and of tin must be melted with time is it now? Ans: 21:00
100 lb of the given alloy to produce another alloy 9. Between 3:00 and 4:00, Kathrine looked at her watch and
analyzing 30% copper and 10% tin? All percentages are noticed that the minute hand was between 5 and 6.
by weight. Answer: 17.5 lb copper & 7.5 lb tin Later, Kathrine looked again and noticed that the hour
6. In what ratio must tea costing P240 per kg be mixed with hand and the minute hand had exchanged places. What
tea costing P340 per kg so that a profit of 20% is made by time was it in the second case? Answer: 5:17.20
selling the mixture at P360 per kg? Answer: 2:3 10. A businessman left his office for an appointment and
noticed the hands of the wall clock at past 8 o’clock. After
more than two hours he returned and noticed that the
Proportion Problems: hands of the clock were interchanged. At what time did
7. Albert, Bryan and Carl are partners who share profit in he leave and at what time did he return?
the ratio of their capital. Albert’s capital is Answer: leave = 8:58:44; return = 11:44:53
P5M and Bryan's is P25M. In 2018, the profit was P3.5M,
11. What percentage of a mixture of sand, gravel, and
in which Carl's share was P1.5M. Find Carl's capital.
Answer: P22.5M
cement containing 30% cement should be replaced by
pure cement to produce a mixture that is 40% cement.
8. On a fence are sparrows and pigeons. When 5 sparrows Answer: 14.3%
leave, there remain two pigeons for every sparrow. Then
12. How much water must be evaporated from a 15-liter 12%
twenty-five pigeons leave, and there now three sparrows
dye solution to obtain a solution that is 20% dye?
for every pigeon. Find the original number of sparrows.
Answer: 20
Assume that the total amount of dye is not affected by
the process of evaporation. Answer: 6 liters
13. A radiator has 25 liters of 30 percent antifreeze solution.
Variation Problems: How much of this should be drained and then replaced
x varies directly as y with pure antifreeze material to make 45% antifreeze?
® x µ y or x = ky Answer: 5.36 liters
x varies inversely as y
14. A chemist has three different acid solutions. The first
1 æ1ö acid solution contains 15% acid, the second contains 30%
®xµ or x = k ç ÷
y èyø acid, and the third contains 75% acid. He wants to use
x varies jointly as y and z all three solutions to obtain a mixture of 216 liters
® x µ yz or x = kyz containing 25% acid, using 2 times as much of the 75%
solution as the 30% solution. How many liters of the 15%
9. The electrical resistance of a cable varies directly as its acid solution should be used? Ans: 168 liters
length and inversely as the square of its diameter. If a 15. Write the variation equation described as follows: The
cable 600m long and 10mm in diameter has a resistance force of electrical attraction (Fel) between two statically
of 0.1 ohm, find the length of a cable 20 mm in diameter charged bodies is directly proportional to the product of
with a resistance of 0.15 ohm. Answer: 3600 m the charges on each of the two objects (q1, q2) and
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Review Innovations CE Review for Nov 2022 – Algebra 6
inversely proportional to the square of the distance (d)
separating these two charged bodies.
16. X and Y are inversely proportional with each other.
Given that X = 15,000 when Y = 162,500. Find X when Y
= 328,400. Answer: 7422.35
17. According to Hooke’s law, the force F required to stretch
a spring by x units is directly proportional to x. If a force
of 20 kN is needed to stretch a spring by 3 mm, how far
is the spring stretched by a force of 60 kN?
Ans: 9 mm
18. According to Boyle’s law, the pressure p of a compressed
gas is inversely proportional to the volume V. Suppose
that there is a pressure of 25 kPa when the volume of the
gas is 400 cu. cm. Find the pressure when the gas is
compressed to 200 cu. cm. Answer: 50 kPa
19. The pressure P of wind on a sail varies jointly as the area
A of the sail and the square of the velocity V of the wind.
The pressure on a square foot is 1 lb when the velocity is
16 mph. The velocity of the wind when the pressure on
a square yard is 36 lbs is: Answer: 32 mph
20. When looking at two buildings at the same time, the
length of the buildings’ shadows, s, varies directly as
their height, h. If a 5-story building has a 20 m long
shadow, how many stories high would a building that
has a 32 m long shadow be? Answer: 8 stories
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