If there is one prayer that you should pray/sing every day and every hour, it is the
LORD's prayer (Our FATHER in Heaven prayer)
- Samuel Dominic Chukwuemeka
It is the most powerful prayer.
A pure heart, a clean mind, and a clear conscience is necessary for it.
For in GOD we live, and move, and have our being.
- Acts 17:28
The Joy of a Teacher is the Success of his Students.
- Samuel Dominic Chukwuemeka
For ACT Students
The ACT is a timed exam...60 questions for 60 minutes
This implies that you have to solve each question in one minute.
Some questions will typically take less than a minute a solve.
Some questions will typically take more than a minute to solve.
The goal is to maximize your time. You use the time saved on those questions you
solved in less than a minute, to solve the questions that will take more than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no negative penalty for any wrong answer.
For JAMB Students
Calculators are not allowed. So, the questions are solved in a way that does not require a calculator.
For WASSCE Students
Any question labeled WASCCE is a question for the WASCCE General Mathematics
Any question labeled WASSCE-FM is a question for the WASSCE Further Mathematics/Elective Mathematics
For NSC Students
For the Questions:
Any space included in a number indicates a comma used to separate digits...separating multiples of three digits
from behind.
Any comma included in a number indicates a decimal point.
For the Solutions:
Decimals are used appropriately rather than commas
Commas are used to separate digits appropriately.
Solve all questions.
Show all work.
Unless specified or implied otherwise, leave all applicable answers as simplified fractions or integers.
NOTE:
(1.) For all the questions:
S is the sample space
n(S) is the cardinality of the sample space.
(2.) In some cases, it is recommended that you do not simplify until the final answer.
(3.) Prerequisites:
(a.) Fractions, Decimals, and Percents
(b.) Expressions and Equations
(c.) Set Algebra for some questions.
(4.) Corequisites:
(a.) Combinatorics
Some of these questions are preferably solved using Combinatorics.
You can review those same questions solved using Combinatorics
(b.) Binomial Distribution Applications
Some of these questions may be solved using Binomial Probability Distribution.
Woman → Man ↓ |
$A$ | $S$ |
---|---|---|
$A$ | $AA$ | $AS$ |
$S$ | $AS$ | $SS$ |
Woman → Man ↓ |
$A$ | $S$ |
---|---|---|
$A$ | $AA$ | $AS$ |
$S$ | $AS$ | $SS$ |
21033 17516 39329 45506 |
32522 69328 17705 44716 |
19305 88389 70251 02498 |
90633 19770 84413 15327 |
80873 33197 30896 79149 |
19167 27336 21490 28409 |
21033 17516 26613 45506 |
32522 69328 04566 44716 |
19305 88389 24986 02498 |
90633 19770 78498 15327 |
80873 33197 30896 79149 |
19167 27336 21490 28409 |
$(+)$ | $2$ | $3$ | $4$ |
$1$ | $3$ | $\color{red}{4}$ | $\color{red}{5}$ |
$3$ | $\color{red}{5}$ | $\color{red}{6}$ | $7$ |
$5$ | $7$ | $8$ | $9$ |
Scores | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
Frequency | $25$ | $30$ | $x$ | $28$ | $40$ | $32$ |
10th Grade | 11th Grade | 12th Grade | Total | |
Math Guys | $232$ | $442$ | $59$ | $733$ |
Art Guys | $1036$ | $68$ | $53$ | $1157$ |
Total | $1268$ | $510$ | $112$ | $1890$ |
Type of chip | Number of chips made by Company: | |
$A$ | $B$ | |
$I$ $II$ |
$14$ $36$ |
$30$ $20$ |
True | False | |
---|---|---|
Positive Result | $120$ | $26$ |
Negative Result | $157$ | $3$ |
Hours of Sleep | Number of Citizens (in millions) |
---|---|
$4\:\:or\:\:less$ | $11$ |
$5$ | $31$ |
$6$ | $74$ |
$7$ | $89$ |
$8$ | $81$ |
$9$ | $10$ |
$10\:\:or\:\:more$ | $4$ |
Color | Number |
---|---|
Blue Red Black White Green |
$10$ $8$ $6$ $4$ $2$ |
Color | Probability |
---|---|
red | $0.25$ |
green | $0.2$ |
blue | $0.15$ |
brown | $0$ |
yellow | $0.35$ |
orange | $0.1$ |
Value of Die | Frequency, $F$ |
---|---|
$1$ | $71$ |
$2$ | $62$ |
$3$ | $66$ |
$4$ | $63$ |
$5$ | $70$ |
$6$ | $68$ |
$\Sigma F = 400$ |
Value of Die | Frequency, $F$ |
---|---|
$1$ | $43$ |
$2$ | $100$ |
$3$ | $44$ |
$4$ | $114$ |
$5$ | $50$ |
$6$ | $49$ |
$\Sigma F = 400$ |
$(+)$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
$2$ | $3$ | $4$ | $5$ | $6$ | $\color{red}{7}$ |
$3$ | $4$ | $5$ | $6$ | $\color{red}{7}$ | $8$ |
$4$ | $5$ | $6$ | $\color{red}{7}$ | $8$ | $9$ |
$5$ | $6$ | $\color{red}{7}$ | $8$ | $9$ | $10$ |
Fielding position | Frequency |
Catcher Infielder Pitcher Outfielder |
$4$ $6$ $8$ $7$ |
Student status | Approve | Disapprove | No opinion |
High school College Nonstudent |
$30$ $14$ $85$ |
$4$ $10$ $353$ |
$11$ $6$ $47$ |
Total | $129$ | $367$ | $64$ |
$29$ | $38$ | $26$ | $42$ | $38$ |
$45$ | $35$ | $37$ | $38$ | $31$ |
$1\:\:Coin\:\:\rightarrow$ $1\:\:Die\:\:\downarrow$ |
$H$ | $T$ |
---|---|---|
$1$ | $1H$ | $1T$ |
$2$ | $2H$ | $2T$ |
$3$ | $3H$ | $3T$ |
$4$ | $4H$ | $4T$ |
$5$ | $5H$ | $5T$ |
$6$ | $6H$ | $6T$ |
Color | Quantity |
Green Black Red Orange Yellow Blue |
$6$ $3$ $5$ $2$ $4$ $8$ |
Order | Decaf | Regular | Total |
With milk Without milk |
$12$ $6$ |
$8$ $10$ |
$20$ $16$ |
Total | $18$ | $18$ | $36$ |
$y\downarrow\:\:(+)\:\:x\rightarrow$ | $1$ | $2$ | $3$ | $4$ | $5$ |
$1$ | $2$ | $\color{red}{3}$ | $4$ | $\color{red}{5}$ | $6$ |
$2$ | $\color{red}{3}$ | $4$ | $\color{red}{5}$ | $6$ | $\color{red}{7}$ |
$3$ | $4$ | $\color{red}{5}$ | $6$ | $\color{red}{7}$ | $8$ |
$2\:\:Flips\:\:\rightarrow$ $2\:\:Flips\:\:\downarrow$ |
$HH$ | $HT$ | $TH$ | $TT$ |
---|---|---|---|---|
$HH$ | $HHHH$ | $HHHT$ | $HHTH$ | $HHTT$ |
$HT$ | $HTHH$ | $HTHT$ | $HTTH$ | $HTTT$ |
$TH$ | $THHH$ | $THHT$ | $THTH$ | $THTT$ |
$TT$ | $TTHH$ | $TTHT$ | $TTTH$ | $TTTT$ |
$1\:\:Die\:\:\rightarrow$ ($+$) $1\:\:Die\:\:\downarrow$ |
$1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
---|---|---|---|---|---|---|
$1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ |
$2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
$3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ |
$4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
$5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ |
$6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ |
$1\:\:Die\:\:\rightarrow$ ($+$) $1\:\:Die\:\:\downarrow$ |
$1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
---|---|---|---|---|---|---|
$1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ |
$2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
$3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ |
$4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
$5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ |
$6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ |
$1\:\:Die\:\:\rightarrow$ ($+$) $1\:\:Die\:\:\downarrow$ |
$1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
---|---|---|---|---|---|---|
$1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ |
$2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
$3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ |
$4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
$5$ | $6$ | $7$ | $8$ | $9$ | $10$ | $11$ |
$6$ | $7$ | $8$ | $9$ | $10$ | $11$ | $12$ |
$Square\;\;numbers\:\:\rightarrow$ ($+$) $Prime\;\;numbers\:\:\downarrow$ |
$1$ | $4$ | $9$ | $16$ |
---|---|---|---|---|
$2$ | $3$ | $6$ | $11$ | $18$ |
$3$ | $4$ | $7$ | $12$ | $19$ |
$5$ | $6$ | $9$ | $14$ | $21$ |
$7$ | $8$ | $11$ | $16$ | $23$ |
1 | 2 | 3 | 4 |
5 | 6 | 7 | 8 |
$ Rectangular\;\;Table\:\:\rightarrow \\[1ex] \;\;\;\;\;* \\[1ex] Standard\;\;Die\:\:\downarrow $ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
---|---|---|---|---|---|---|---|---|
$1$ | $1$ | $2$ | $3$ | $4$ | $5$ | $\color{red}6$ | $7$ | $8$ |
$2$ | $2$ | $4$ | $\color{red}6$ | $8$ | $10$ | $12$ | $14$ | $16$ |
$3$ | $3$ | $\color{red}6$ | $9$ | $12$ | $15$ | $18$ | $21$ | $24$ |
$4$ | $4$ | $8$ | $12$ | $16$ | $20$ | $24$ | $28$ | $32$ |
$5$ | $5$ | $10$ | $15$ | $20$ | $25$ | $30$ | $35$ | $40$ |
$6$ | $\color{red}6$ | $12$ | $18$ | $24$ | $30$ | $36$ | $42$ | $48$ |
$1\:\:Toss\:\:\rightarrow$ ($+$) $1\:\:Toss\:\:\downarrow$ |
$1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
---|---|---|---|---|---|---|
$1$ | $2$ | $\color{red}{3}$ | $\color{violet}{4}$ | $\color{violet}{5}$ | $\color{red}{6}$ | $7$ |
$2$ | $\color{red}{3}$ | $\color{violet}{4}$ | $\color{violet}{5}$ | $\color{red}{6}$ | $7$ | $8$ |
$3$ | $\color{violet}{4}$ | $\color{violet}{5}$ | $\color{red}{6}$ | $7$ | $8$ | $\color{red}{9}$ |
$4$ | $\color{violet}{5}$ | $\color{red}{6}$ | $7$ | $8$ | $\color{red}{9}$ | $10$ |
$5$ | $\color{red}{6}$ | $7$ | $8$ | $\color{red}{9}$ | $10$ | $11$ |
$6$ | $7$ | $8$ | $\color{red}{9}$ | $10$ | $11$ | $\color{red}{12}$ |
$Panel\;2\:\:\rightarrow$ ($+$) $Panel\:1\:\:\downarrow$ |
$1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
---|---|---|---|---|---|---|
$1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $\color{red}{7}$ |
$2$ | $3$ | $4$ | $5$ | $6$ | $\color{red}{7}$ | $\color{red}{8}$ |
$3$ | $4$ | $5$ | $6$ | $\color{red}{7}$ | $\color{red}{8}$ | $\color{red}{9}$ |
$1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $\color{red}{7}$ |
$1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $\color{red}{7}$ |
$1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $\color{red}{7}$ |
$3$ | $4$ | $5$ | $6$ | $\color{red}{7}$ | $\color{red}{8}$ | $\color{red}{9}$ |
$2$ | $3$ | $4$ | $5$ | $6$ | $\color{red}{7}$ | $\color{red}{8}$ |
$2nd\:\:Roll\:\:\rightarrow$ ($+$) $1st\:\:Roll\:\:\downarrow$ |
$1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
---|---|---|---|---|---|---|---|---|
$1$ | $1, 1$ | $1, 2$ | $1, 3$ | $1, 4$ | $1, 5$ | $1, 6$ | $1, 7$ | $1, 8$ |
$2$ | $2, 1$ | $2, 2$ | $2, 3$ | $2, 4$ | $2, 5$ | $2, 6$ | $2, 7$ | $2, 8$ |
$3$ | $3, 1$ | $3, 2$ | $3, 3$ | $3, 4$ | $3, 5$ | $3, 6$ | $3, 7$ | $3, 8$ |
$4$ | $4, 1$ | $4, 2$ | $4, 3$ | $4, 4$ | $4, 5$ | $4, 6$ | $4, 7$ | $4, 8$ |
$5$ | $5, 1$ | $5, 2$ | $5, 3$ | $5, 4$ | $5, 5$ | $5, 6$ | $5, 7$ | $5, 8$ |
$6$ | $6, 1$ | $6, 2$ | $6, 3$ | $6, 4$ | $6, 5$ | $6, 6$ | $6, 7$ | $6, 8$ |
$7$ | $7, 1$ | $7, 2$ | $7, 3$ | $7, 4$ | $7, 5$ | $7, 6$ | $7, 7$ | $7, 8$ |
$8$ | $8, 1$ | $8, 2$ | $8, 3$ | $8, 4$ | $8, 5$ | $8, 6$ | $8, 7$ | $8, 8$ |
Number | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
Outcomes | $32$ | $m$ | $25$ | $40$ | $28$ | $45$ |
$2nd\:\:Number\:\:\rightarrow$ $1st\:\:Number\:\:\downarrow$ |
$5$ | $6$ | $7$ | $8$ | $9$ |
---|---|---|---|---|---|
$5$ | $\color{red}{5, 5}$ | $\color{red}{5, 6}$ | $\color{red}{5, 7}$ | $\color{red}{5, 8}$ | $\color{red}{5, 9}$ |
$6$ | $\color{red}{6, 5}$ | $6, 6$ | $\color{red}{6, 7}$ | $6, 8$ | $6, 9$ |
$7$ | $\color{red}{7, 5}$ | $\color{red}{7, 6}$ | $\color{red}{7, 7}$ | $\color{red}{7, 8}$ | $\color{red}{7, 9}$ |
$8$ | $\color{red}{8, 5}$ | $8, 6$ | $\color{red}{8, 7}$ | $8, 8$ | $8, 9$ |
$9$ | $\color{red}{9, 5}$ | $9, 6$ | $\color{red}{9, 7}$ | $9, 8$ | $9, 9$ |
$ Die\:\:1\;\rightarrow \\[1ex] \;\;\;- \\[1ex] Die\:\:2\;\downarrow $ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
---|---|---|---|---|---|---|
$1$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ |
$2$ | $\color{red}1$ | $0$ | $1$ | $2$ | $3$ | $4$ |
$3$ | $2$ | $1$ | $\color{red}0$ | $\color{red}1$ | $2$ | $\color{red}3$ |
$4$ | $3$ | $2$ | $1$ | $\color{red}0$ | $1$ | $2$ |
$5$ | $\color{red}4$ | $3$ | $2$ | $1$ | $0$ | $1$ |
$6$ | $5$ | $4$ | $3$ | $2$ | $1$ | $0$ |
$2nd\:\:Throw\:\:\rightarrow$ ($+$) $1st\:\:Throw\:\:\downarrow$ |
$1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
---|---|---|---|---|---|---|
$1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ |
$2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ |
$3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ |
$4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $\color{red}{10}$ |
$5$ | $6$ | $7$ | $8$ | $9$ | $\color{red}{10}$ | $\color{red}{11}$ |
$6$ | $7$ | $8$ | $9$ | $\color{red}{10}$ | $\color{red}{11}$ | $\color{red}{12}$ |
ACT Use the following information to answer Questions 105 - 108.
Many humans carry the gene Yq77.
The Yq test determines, with 100% accuracy, whether a human carries Yq77.
If a Yq test result is negative, the human does NOT carry Yq77
Sam designed a less expensive test for Yq77 called the Sam77 test.
It produces some incorrect results.
To determine the accuracy of the Sam77 test, both tests were administered to 1,000 volunteers.
The results from this administration are summarized in the table below.
Positive Yq test | Negative Yq test | |
Positive Sam77 test Negative Sam77 test |
590 25 |
10 375 |
North | South | |
Grade 10 Grade 11 |
47 73 |
93 57 |
Grade | Enrollment by May 15 | Regular enrollment fee |
3 4 5 6 |
20 15 28 18 |
$350 $400 $450 $500 |
Balls | $G$ | $R$ | $W$ | $B$ |
Frequency | $2$ | $4$ | $3$ | $1$ |
$2nd\:\:Roll\:\:\rightarrow$ $1st\:\:Roll\:\:\downarrow$ |
$1$ | $2$ | $3$ | $4$ | $5$ | $6$ |
---|---|---|---|---|---|---|
$1$ | $\color{red}{(1, 1)}$ | $(1, 2)$ | $(1, 3)$ | $(1, 4)$ | $(1, 5)$ | $(1, 6)$ |
$2$ | $(2, 1)$ | $(2, 2)$ | $(2, 3)$ | $(2, 4)$ | $(2, 5)$ | $(2, 6)$ |
$3$ | $(3, 1)$ | $(3, 2)$ | $(3, 3)$ | $(3, 4)$ | $(3, 5)$ | $(3, 6)$ |
$4$ | $(4, 1)$ | $(4, 2)$ | $(4, 3)$ | $(4, 4)$ | $(4, 5)$ | $(4, 6)$ |
$5$ | $(5, 1)$ | $(5, 2)$ | $(5, 3)$ | $(5, 4)$ | $(5, 5)$ | $(5, 6)$ |
$6$ | $(6, 1)$ | $(6, 2)$ | $(6, 3)$ | $(6, 4)$ | $(6, 5)$ | $(6, 6)$ |
$2\:\:Tosses\:\:\rightarrow$ $4\:\:Tosses\:\:\downarrow$ |
$HH$ | $HT$ | $TH$ | $TT$ |
---|---|---|---|---|
$HHHH$ | $HHHHHH$ | $HHHHHT$ | $HHHHTH$ | $HHHHTT$ |
$HHHT$ | $HHHTHH$ | $HHHTHT$ | $HHHTTH$ | $HHHTTT$ |
$HHTH$ | $HHTHHH$ | $HHTHHT$ | $HHTHTH$ | $HHTHTT$ |
$HHTT$ | $HHTTHH$ | $HHTTHT$ | $HHTTTH$ | $HHTTTT$ |
$HTHH$ | $HTHHHH$ | $HTHHHT$ | $HTHHTH$ | $HTHHTT$ |
$HTHT$ | $HTHTHH$ | $HTHTHT$ | $HTHTTH$ | $HTHTTT$ |
$HTTH$ | $HTTHHH$ | $HTTHHT$ | $HTTHTH$ | $HTTHTT$ |
$HTTT$ | $HTTTHH$ | $HTTTHT$ | $HTTTTH$ | $HTTTTT$ |
$THHH$ | $THHHHH$ | $THHHHT$ | $THHHTH$ | $THHHTT$ |
$THHT$ | $THHTHH$ | $THHTHT$ | $THHTTH$ | $THHTTT$ |
$THTH$ | $THTHHH$ | $THTHHT$ | $THTHTH$ | $THTHTT$ |
$THTT$ | $THTTHH$ | $THTTHT$ | $THTTTH$ | $THTTTT$ |
$TTHH$ | $TTHHHH$ | $TTHHHT$ | $TTHHTH$ | $TTHHTT$ |
$TTHT$ | $TTHTHH$ | $TTHTHT$ | $TTHTTH$ | $TTHTTT$ |
$TTTH$ | $TTTHHH$ | $TTTHHT$ | $TTTHTH$ | $TTTHTT$ |
$TTTT$ | $TTTTHH$ | $TTTTHT$ | $TTTTTH$ | $TTTTTT$ |
$One\:\:Toss\:\:\rightarrow$ $Two\:\:Tosses\:\:\downarrow$ |
$H$ | $T$ |
---|---|---|
$HH$ | $HHH$ | $HHT$ |
$HT$ | $HTH$ | $\color{red}{HTT}$ |
$TH$ | $THH$ | $\color{red}{THT}$ |
$TT$ | $\color{red}{TTH}$ | $TTT$ |
Car companies | ||||
Age (in years) | A | B | C | Total |
16 - 25 26 - 45 46 - 60 |
16 54 65 |
24 48 23 |
40 53 12 |
80 155 100 |
Total | 135 | 95 | 105 | 335 |
Event | Probability |
---|---|
A $0.20$ |
B $0.40$ |