Number and Numeration Alıştırma Soruları — JAMB UTME — Sayfa 12

Soru 221Soru →

A factory employs $12$ workers to produce $180$ identical wooden chairs in $5$ days, working $8$ hours per day. If $4$ workers are reassigned to another department, how many days will the remaining workers take to produce $210$ such chairs if they work $7$ hours per day?

10\text{ days}

20\text{ days}

15\text{ days}

8\text{ days}

Cevabı ve açıklamayı göster

Cevap: $10\text{ days}$

Cevap

The remaining workers will take

10\text{ days}

to complete the task.

The correct answer is

10\text{ days}

. Using compound proportion, the relationship is given by

\frac{W_1 \times D_1 \times H_1}{C_1} = \frac{W_2 \times D_2 \times H_2}{C_2}

. Substituting the given values

\frac{12 \times 5 \times 8}{180} = \frac{8 \times D_2 \times 7}{210}

yields

\frac{480}{180} = \frac{56 D_2}{210}

, which simplifies to

\frac{8}{3} = \frac{56 D_2}{210}

. Solving for

D_2

gives

D_2 = 10\text{ days}

.

Adım Adım Çözüm

1

Calculate the total man-hours required for the initial production batch.

Total man-hours =

12\text{ workers} \times 5\text{ days} \times 8\text{ hours/day} = 480\text{ man-hours}

.

Determining total labor input needed for

180

chairs.

2

Find the man-hours required per chair.

Man-hours per chair =

\frac{480}{180} = \frac{8}{3}\text{ hours/chair}

.

Establishing the unit rate of work.

3

Calculate the total man-hours required for the new target of

210

chairs.

Total man-hours needed =

210 \times \frac{8}{3} = 560\text{ man-hours}

.

Scaling the unit work rate to the new quantity.

4

Determine the new workforce size and daily labor capacity.

Remaining workers =

12 - 4 = 8\text{ workers}

. Daily man-hours =

8 \times 7 = 56\text{ man-hours/day}

.

Accounting for the reduced workforce and new daily hours.

5

Calculate the number of days required.

Days =

\frac{560\text{ man-hours}}{56\text{ man-hours/day}} = 10\text{ days}

.

Dividing total required work by daily capacity.

Anahtar Kavram

Compound Proportion and Work-Rate Relationships

Tahmini Süre:1m 30s

Soru 222Soru →

Given the matrix $A = \begin{pmatrix} 3 & 1 \\ 4 & 2 \end{pmatrix}$ , what is the determinant of the matrix $3A$ ?

6

18

30

54

Cevabı ve açıklamayı göster

Cevap: $18$

Cevap

The determinant of the matrix

3A

is

18

.

For any

2 \times 2

matrix, scaling the matrix by a factor

k

scales its determinant by

k^2

. Since

\det(A) = (3)(2) - (1)(4) = 2

, multiplying the matrix by

3

yields

\det(3A) = 3^2 \times 2 = 9 \times 2 = 18

. Alternatively, computing

3A = \begin{pmatrix} 9 & 3 \\ 12 & 6 \end{pmatrix}

directly gives

\det(3A) = (9)(6) - (3)(12) = 54 - 36 = 18

.

Adım Adım Çözüm

1

Calculate the determinant of matrix

A

.

\det(A) = (3)(2) - (1)(4) = 6 - 4 = 2.

The determinant of a

2 \times 2

matrix

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

is given by

ad - bc

.

2

Apply the determinant scalar multiplication property for an

n \times n

matrix.

\det(3A) = 3^2 \det(A) = 9 \times 2 = 18.

For any

n \times n

square matrix

A

and scalar

k

, the identity

\det(kA) = k^n \det(A)

holds. Here,

k=3

and

n=2

.

Anahtar Kavram

Scalar Multiplication Property of Determinants

Tahmini Süre:1m 15s

Soru 223Soru →

Given the matrix $A = \begin{pmatrix} y & 5 \\ 3 & y+2 \end{pmatrix}$ , if the determinant of $A$ is $9$ and $y > 0$ , calculate the value of $y$ .

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The value of

y

is 4.

Expanding the determinant of

A

yields

\det(A) = y(y+2) - 15 = y^2 + 2y - 15

. Equating this to 9 gives

y^2 + 2y - 24 = 0

, which factors as

(y+6)(y-4) = 0

. The roots are

y = -6

and

y = 4

. Given that

y > 0

, the required value is 4.

Adım Adım Çözüm

1

Find the expression for the determinant of matrix

A

\det(A) = y(y+2) - (5)(3) = y^2 + 2y - 15

The determinant of a

2 \times 2

matrix

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

is given by

ad - bc

.

2

Form and solve the quadratic equation

y^2 + 2y - 15 = 9 \implies y^2 + 2y - 24 = 0 \implies (y + 6)(y - 4) = 0

Set the determinant expression equal to the given determinant value of 9 and rearrange into standard quadratic form.

3

Apply the given domain restriction

y > 0

y = 4

The root

y = -6

is discarded because

y

must be strictly positive.

Anahtar Kavram

Determinant of a 2x2 matrix and solving non-linear determinant equations

Soru 224Soru →

At a regional agricultural exhibition, $150$ farmers registered their crop cultivation. $85$ farmers grow cassava ( $C$ ), $70$ grow maize ( $M$ ), and $60$ grow yam ( $Y$ ). $35$ farmers grow both cassava and maize, $25$ grow both maize and yam, and $30$ grow both cassava and yam. If $15$ farmers grow all three crops, how many farmers grow none of these three crops?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The number of farmers who grow none of the three crops is 10.

Using the Principle of Inclusion-Exclusion for three sets, the number of farmers growing at least one of cassava, maize, or yam is given by

85 + 70 + 60 - 35 - 25 - 30 + 15 = 140

. Since there are

150

farmers in total, the number of farmers who grow none of these three crops is

150 - 140 = 10

.

Adım Adım Çözüm

1

Calculate the cardinality of the union of the three sets using the Principle of Inclusion-Exclusion.

|C \cup M \cup Y| = 85 + 70 + 60 - 35 - 25 - 30 + 15 = 140

Summing the three individual set counts overcounts elements in pairwise intersections, which must be subtracted. The central triple intersection is then added back because it was subtracted once too often.

2

Subtract the size of the union from the size of the universal set.

|(C \cup M \cup Y)'| = 150 - 140 = 10

The complement of the union represents the set of farmers outside all three crop categories.

Anahtar Kavram

Principle of Inclusion-Exclusion for Three Sets

Soru 225Soru →

Given the matrices $M = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}$ and $N = \begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix}$ , what is the determinant of the matrix product $MN$ ?

14

15

24

33

Cevabı ve açıklamayı göster

Cevap: 33

Cevap

33

Using the property that the determinant of a product of matrices equals the product of their individual determinants,

\det(M) = (2)(4) - (-1)(3) = 11

and

\det(N) = (1)(3) - (0)(2) = 3

. Multiplying these gives

\det(MN) = 11 \times 3 = 33

.

Adım Adım Çözüm

1

Calculate the determinant of matrix

M

\det(M) = (2)(4) - (-1)(3) = 8 + 3 = 11

The determinant of a

2 \times 2

matrix

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

is given by

ad - bc

.

2

Calculate the determinant of matrix

N

\det(N) = (1)(3) - (0)(2) = 3 - 0 = 3

Apply the determinant formula

ad - bc

to matrix

N

.

3

Apply the determinant product property to find

\det(MN)

\det(MN) = \det(M) \times \det(N) = 11 \times 3 = 33

For any two square matrices of the same order,

\det(MN) = \det(M) \cdot \det(N)

.

Anahtar Kavram

Determinant Property of Matrix Products

Tahmini Süre:1m 30s

Soru 226Soru →

A poultry farmer in Ogun State took a loan of $\text{₦}120,000$ from a cooperative society at an annual interest rate of $5\%$ , compounded annually. What is the total interest paid by the farmer at the end of $2$ years?

\text{₦}12,300

\text{₦}132,300

\text{₦}12,000

\text{₦}132,000

Cevabı ve açıklamayı göster

Cevap: $\text{₦}12,300$

Cevap

The total compound interest paid at the end of 2 years is

\text{₦}12,300

.

The total compound interest is determined by calculating the accumulated amount

A = 120,000 \times (1.05)^2 = \text{₦}132,300

and subtracting the original principal of

\text{₦}120,000

, yielding

\text{₦}12,300

.

Adım Adım Çözüm

1

Identify the given values for principal, rate, and time

Principal

P = \text{₦}120,000

, Rate

r = 5\% = 0.05

, Time

t = 2\text{ years}

These parameters are required to substitute into the compound amount formula.

2

Calculate the total accumulated amount

A

after 2 years

A = P(1 + r)^t = 120,000 \times (1 + 0.05)^2 = 120,000 \times 1.1025 = \text{₦}132,300

The compound interest formula yields the total balance including the initial principal.

3

Subtract the initial principal from the total accumulated amount to determine interest

Compound Interest

= A - P = 132,300 - 120,000 = \text{₦}12,300

Interest is the extra amount generated beyond the original loan amount.

Anahtar Kavram

Compound Interest vs Total Accumulated Amount

Tahmini Süre:1m 30s

Soru 227Soru →

A boutique owner in Enugu borrowed $\text{₦}80,000$ to expand her business at an annual interest rate of $15\%$ , compounded annually. If she repays the loan in full after $2$ years, what is the total interest she paid on the loan?

\text{₦}24,000

\text{₦}25,800

\text{₦}27,600

\text{₦}105,800

Cevabı ve açıklamayı göster

Cevap: $\text{₦}25,800$

Cevap

The total interest paid on the loan after 2 years is

\text{₦}25,800

.

The compound interest is calculated year by year. In the first year, the interest paid is

15\%

of

\text{₦}80,000 = \text{₦}12,000

, raising the balance to

\text{₦}92,000

. In the second year, the interest paid is

15\%

of

\text{₦}92,000 = \text{₦}13,800

. Adding the two yearly interest payments gives

\text{₦}12,000 + \text{₦}13,800 = \text{₦}25,800

.

Adım Adım Çözüm

1

Calculate the interest for the first year.

Interest for Year 1 =

15\%

of

\text{₦}80,000 = \frac{15}{100} \times 80,000 = \text{₦}12,000

.

In compound interest, interest for the first period is calculated on the initial principal.

2

Determine the amount at the end of the first year, which becomes the principal for the second year.

Principal for Year 2 =

\text{₦}80,000 + \text{₦}12,000 = \text{₦}92,000

.

Compound interest adds earned interest to the principal for subsequent period calculations.

3

Calculate the interest for the second year.

Interest for Year 2 =

15\%

of

\text{₦}92,000 = \frac{15}{100} \times 92,000 = \text{₦}13,800

.

Interest in year 2 is computed on the updated principal balance of

\text{₦}92,000

.

4

Sum the interest amounts from both years to find total interest paid.

Total Interest =

\text{₦}12,000 + \text{₦}13,800 = \text{₦}25,800

.

The total interest is the sum of interest accumulated across each compounding period.

Anahtar Kavram

Compound Interest Calculation

Tahmini Süre:1m 30s

Soru 228Soru →

A cooperative society in Akure granted a loan of $\text{₦}160,000$ to a farmer at an interest rate of $12\%$ per annum, compounded annually. What is the total compound interest owed at the end of $2$ years?

\text{₦}40,704

\text{₦}200,704

\text{₦}38,400

\text{₦}19,200

Cevabı ve açıklamayı göster

Cevap: $\text{₦}40,704$

Cevap

The total compound interest owed at the end of 2 years is

\text{₦}40,704

.

The compound interest is obtained by calculating the accumulated amount

A = 160,000 \times (1.12)^2 = \text{₦}200,704

and then subtracting the principal of

\text{₦}160,000

to get

\text{₦}40,704

.

Adım Adım Çözüm

1

Identify the given financial variables

Principal

P = \text{₦}160,000

, Rate

R = 12\%

, Time

n = 2

years.

Establishing the parameters needed for the compound interest formula.

2

Calculate the total accumulated amount

A

using the formula

A = P\left(1 + \frac{R}{100}\right)^n

A = 160,000 \times \left(1 + 0.12\right)^2 = 160,000 \times (1.12)^2 = 160,000 \times 1.2544 = \text{₦}200,704

.

Determines the total value of the loan including principal and accrued compound interest after 2 years.

3

Subtract the principal

P

from the total amount

A

to find the compound interest

CI

CI = A - P = 200,704 - 160,000 = \text{₦}40,704

.

Isolates the interest portion from the total accumulated balance.

Anahtar Kavram

Compound Interest Calculation

Soru 229Soru →

If the matrix $P = \begin{pmatrix} k & 2 & 0 \\ 1 & 4 & 1 \\ 3 & 0 & 2 \end{pmatrix}$ has a determinant equal to $18$ , find the value of $k$ .

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

k

is

2

.

Expanding matrix

P

along the first row gives

\det(P) = k(8 - 0) - 2(2 - 3) + 0(0 - 12) = 8k + 2

. Setting

8k + 2 = 18

yields

8k = 16

, which simplifies to

k = 2

.

Adım Adım Çözüm

1

Expand the matrix determinant along the first row.

\det(P) = k(8 - 0) - 2(2 - 3) + 0 = 8k + 2

Applying the cofactor expansion formula for a

3 \times 3

matrix across row 1.

2

Set the resulting determinant expression equal to the given value of 18.

8k + 2 = 18

The question specifies that the determinant of

P

is equal to

18

.

3

Solve the linear equation for

k

.

k = 2

Subtracting 2 from both sides yields

8k = 16

, and dividing by 8 gives

k = 2

.

Anahtar Kavram

Determinant of a

3 \times 3

Matrix

Number and Numeration

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