Tüm alıştırma soruları

Soru 5041Soru →

A tailor has a bolt of fabric that is $20\frac{1}{4}$ yards long. He cuts $5\frac{3}{4}$ yards of fabric to make a suit jacket and another $3\frac{5}{8}$ yards to make a pair of trousers. How many yards of fabric are remaining on the bolt?

10\frac{7}{8}

11\frac{7}{12}

18\frac{1}{8}

11\frac{1}{8}

Cevabı ve açıklamayı göster

Cevap: $10\frac{7}{8}$

Cevap

10\frac{7}{8}

The correct answer is found by first calculating the total fabric used (

5\frac{3}{4} + 3\frac{5}{8} = 9\frac{3}{8}

yards). Subtracting this from the original

20\frac{1}{4}

yards requires converting to a common denominator (

20\frac{2}{8} - 9\frac{3}{8}

) and then borrowing from the whole number (

19\frac{10}{8} - 9\frac{3}{8}

) to get exactly

10\frac{7}{8}

yards.

Adım Adım Çözüm

1

Find the total amount of fabric used by adding

5\frac{3}{4}

and

3\frac{5}{8}

.

5\frac{3}{4} + 3\frac{5}{8} = 5\frac{6}{8} + 3\frac{5}{8} = 8\frac{11}{8} = 9\frac{3}{8}

yards.

Before finding the remaining amount, the total amount removed from the bolt must be calculated by finding a common denominator.

2

Subtract the total used fabric from the original length of the bolt.

20\frac{1}{4} - 9\frac{3}{8} = 20\frac{2}{8} - 9\frac{3}{8}

.

Subtracting the used amount from the initial amount determines how much fabric is left.

3

Borrow from the whole number to complete the subtraction.

19\frac{10}{8} - 9\frac{3}{8} = 10\frac{7}{8}

yards.

Since

2

is less than

3

in the numerators,

1

whole (

8/8

) must be borrowed from

20

to make the subtraction possible.

Anahtar Kavram

Adding and subtracting mixed numbers with unlike denominators, including borrowing.

Soru 5042Soru →

Evaluate the following mathematical expression:

\frac{7}{8} - \frac{2}{3} \times 1\frac{1}{4}

Which of the following is the correct result?

\frac{25}{96}

1

\frac{1}{24}

-\frac{1}{24}

Cevabı ve açıklamayı göster

Cevap: $\frac{1}{24}$

Cevap

The correct result is

\frac{1}{24}

.

The correct answer is found by first performing the multiplication (resulting in

\frac{5}{6}

) and then subtracting this from

\frac{7}{8}

using a common denominator of 24.

Adım Adım Çözüm

1

Convert the mixed number to an improper fraction.

1\frac{1}{4} = \frac{5}{4}

Fractions must be in proper or improper form before multiplying.

2

Multiply the fractions according to the order of operations (PEMDAS).

\frac{2}{3} \times \frac{5}{4} = \frac{10}{12}

Multiplication must be performed before subtraction in mathematical expressions.

3

Simplify the resulting fraction.

\frac{10}{12} = \frac{5}{6}

Simplifying makes finding a common denominator easier in the next step.

4

Find a common denominator and subtract the fractions.

\frac{7}{8} - \frac{5}{6} = \frac{21}{24} - \frac{20}{24} = \frac{1}{24}

Fractions require a common denominator before they can be added or subtracted.

Anahtar Kavram

Order of Operations with Fractions

Soru 5043Soru →

When the expression $(3x - 4)(2x + 5) - x(x - 2)$ is fully expanded and simplified into standard polynomial form, what is the value of the coefficient of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

9

Expanding the binomials gives

6x^2 + 7x - 20

. Distributing

-x

gives

-x^2 + 2x

. Combining like terms results in

5x^2 + 9x - 20

. The question asks for the coefficient of

x

, which is the number multiplying the

x

term: 9.

Adım Adım Çözüm

1

Multiply the binomials

(3x - 4)

and

(2x + 5)

.

6x^2 + 15x - 8x - 20

, which simplifies to

6x^2 + 7x - 20

.

To expand the first part of the expression using the distributive property (FOIL).

2

Distribute

-x

across the binomial

(x - 2)

.

-x^2 + 2x

.

To expand the second part of the expression, ensuring the negative sign is applied to both terms inside the parentheses.

3

Combine the expanded parts.

(6x^2 + 7x - 20) + (-x^2 + 2x) = 5x^2 + 9x - 20

.

To simplify the full expression into standard polynomial form by grouping like terms.

4

Identify the coefficient of

x

.

The coefficient of

x

in

5x^2 + 9x - 20

is

9

.

This answers the specific question asked.

Anahtar Kavram

Multiplying polynomials and combining like terms

Soru 5044Soru →

Consider the algebraic expression $\frac{2x^2 - yz}{x + y}$ . If the variables are assigned the values $x = -3$ , $y = 5$ , and $z = -2$ , what are the intermediate evaluated values for the numerator and denominator, and what is the final result?

Aşağıdaki boşlukları doldurun

After substituting the values, the evaluated result of the numerator is and the denominator is . Dividing these intermediate values yields a final result of .

Cevabı ve açıklamayı göster

Cevap

The intermediate numerator evaluates to 28, the denominator to 2, and the final evaluated result is 14.

The correct final value is 14. This is determined by carefully substituting the negative and positive variable values into the numerator and denominator, strictly applying the order of operations to handle exponents and negative signs, and finally dividing the intermediate numerator 28 by the denominator 2.

Adım Adım Çözüm

1

Substitute the value of

x

into the first term of the numerator,

2x^2

.

2(-3)^2 = 2(9) = 18

Applying the order of operations, exponentiation must be evaluated before multiplication.

2

Substitute the values of

y

and

z

into the second term of the numerator,

yz

.

(5)(-2) = -10

Multiplying a positive integer by a negative integer results in a negative integer.

3

Calculate the complete numerator by subtracting the second term from the first.

18 - (-10) = 18 + 10 = 28

Subtracting a negative number is mathematically equivalent to adding its positive counterpart.

4

Substitute the values of

x

and

y

to evaluate the denominator,

x + y

.

-3 + 5 = 2

Combining the negative and positive integers.

5

Divide the fully evaluated numerator by the evaluated denominator.

\frac{28}{2} = 14

The final step in evaluating the rational expression.

Anahtar Kavram

Evaluating Algebraic Expressions

Soru 5045Soru →

What is the exact value of the expression $-4.2 + 3.6 \times (1.5 - 2.7)$ ?

-8.52

0.72

0.12

-4.74

Cevabı ve açıklamayı göster

Cevap: $-8.52$

Cevap

The correct value is

-8.52

.

The correct answer follows the standard order of operations and proper arithmetic rules for decimals and integers. First, the subtraction inside the parentheses is calculated as

1.5 - 2.7 = -1.2

. Next, the multiplication is performed before the addition:

3.6 \times (-1.2) = -4.32

. Finally,

-4.32

is added to

-4.2

, making sure to properly align the decimal points, which yields

-8.52

.

Adım Adım Çözüm

1

Evaluate the expression inside the parentheses.

1.5 - 2.7 = -1.2

According to the order of operations (PEMDAS), operations inside parentheses must be completed first.

2

Multiply the result from step 1 by

3.6

.

3.6 \times (-1.2) = -4.32

Multiplication takes precedence over addition.

3

Add

-4.2

to the product from step 2.

-4.2 + (-4.32) = -8.52

This is the final remaining operation. Decimal points must be aligned vertically (

-4.20 + (-4.32)

) to ensure correct addition.

Anahtar Kavram

Decimal Operations with Order of Operations

Soru 5046Soru →

In a mathematical model, the output variable $M$ is defined by the equation $M = 2a^2 - 3ab + b^3$ . What is the computed value of $M$ if the input parameters are set to $a = -3$ and $b = -2$ ?

-8

10

28

-28

Cevabı ve açıklamayı göster

Cevap: -8

Cevap

-8

The correct answer is found by substituting the values and strictly following the order of operations. First, the exponents are evaluated:

(-3)^2 = 9

and

(-2)^3 = -8

. Next, the multiplication is performed:

2(9) = 18

and

-3(-3)(-2) = -18

. Finally, combining these values gives

18 - 18 - 8 = -8

.

Adım Adım Çözüm

1

Substitute the values

a = -3

and

b = -2

into the original equation

M = 2a^2 - 3ab + b^3

.

M = 2(-3)^2 - 3(-3)(-2) + (-2)^3

Direct substitution is the first step in evaluating any algebraic expression.

2

Evaluate the terms with exponents according to the order of operations (PEMDAS/BODMAS).

(-3)^2 = 9

and

(-2)^3 = -8

Exponents must be calculated before multiplication. A negative number squared is positive, while a negative number cubed remains negative.

3

Perform the multiplication for each term in the expression.

2(9) = 18

and

-3(-3)(-2) = -18

Multiplication is performed next. For the middle term, multiplying three negative numbers results in a negative product.

4

Combine the resulting values using addition and subtraction.

18 - 18 + (-8) = 0 - 8 = -8

This final arithmetic operation yields the evaluated result of the expression.

Anahtar Kavram

Evaluating algebraic expressions involving exponents, negative bases, and strict adherence to the order of operations.

Tahmini Süre:1m 0s

Soru 5047Soru →

Solve the following linear equation for $w$ :

\frac{1}{3}(12w - 9) + 5 = 2(w - 1) + 8

Enter the exact integer value that satisfies the equation.

Aşağıdaki boşlukları doldurun

The value of

w

is .

Cevabı ve açıklamayı göster

Cevap

2

By successfully distributing the factors, combining like terms on each side, and balancing the equation by moving variables and constants to opposite sides, the isolated variable equates to exactly 2.

Adım Adım Çözüm

1

Distribute the coefficients to remove parentheses on both sides of the equation.

4w - 3 + 5 = 2w - 2 + 8

Applying the distributive property simplifies the expression and exposes the variable terms.

2

Combine the constant terms on each side.

4w + 2 = 2w + 6

Simplifying each side makes it easier to balance the equation in the next steps.

3

Subtract

2w

from both sides of the equation.

2w + 2 = 6

This groups all the variable terms onto the left side.

4

Subtract

2

from both sides, then divide the result by

2

.

2w = 4 \Rightarrow w = 2

This completely isolates the variable

w

to reveal its final value.

Anahtar Kavram

Solving One-Variable Linear Equations

Soru 5048Soru →

A machinist needs to drill a hole completely through a steel block that is $3\frac{3}{4}$ inches thick. The drill bit advances through the steel at a constant rate of $\frac{3}{16}$ of an inch per second. How many seconds will it take to drill completely through the block?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

It will take 20 seconds to drill completely through the block.

To find the time required, the total thickness of the block must be divided by the distance the drill advances per second. Converting

3\frac{3}{4}

to

\frac{15}{4}

and dividing by

\frac{3}{16}

is calculated by multiplying by the reciprocal:

\frac{15}{4} \times \frac{16}{3} = 20

.

Adım Adım Çözüm

1

Convert the mixed number to an improper fraction.

3\frac{3}{4} = \frac{15}{4}

Fraction multiplication and division are performed using improper fractions rather than mixed numbers.

2

Set up the division equation to solve for time.

Time = Distance

\div

Rate =

\frac{15}{4} \div \frac{3}{16}

The time required is equal to the total distance (thickness) divided by the rate of progress.

3

Rewrite the division problem as multiplication by the reciprocal.

\frac{15}{4} \times \frac{16}{3}

Dividing by a fraction is mathematically equivalent to multiplying by its reciprocal.

4

Simplify the fractions by cross-canceling, then multiply.

5 \times 4 = 20

Cross-canceling common factors (15 and 3, 16 and 4) before multiplying makes the final calculation simpler.

Anahtar Kavram

Fraction Division and Applied Rates

Tahmini Süre:50s

Soru 5049Soru →

What is the numerical value of the algebraic expression $-5x^2 + \frac{12y}{x} - y^2$ when evaluated for $x = -2$ and $y = -3$ ?

29

7

-11

109

Cevabı ve açıklamayı göster

Cevap: -11

Cevap

The correct numerical value of the expression is -11.

Substituting

x = -2

and

y = -3

gives

-5(-2)^2 + \frac{12(-3)}{-2} - (-3)^2

. Evaluating the exponents first results in

-5(4) + \frac{-36}{-2} - 9

. Multiplying and dividing yields

-20 + 18 - 9

. Finally, adding and subtracting gives the correct value of

-11

.

Adım Adım Çözüm

1

Substitute the given numeric values into the algebraic expression using parentheses.

-5(-2)^2 + \frac{12(-3)}{-2} - (-3)^2

Using parentheses during substitution prevents sign errors, especially when variables have negative values.

2

Evaluate the terms with exponents.

-5(4) + \frac{12(-3)}{-2} - (9)

According to the order of operations (PEMDAS), exponents must be evaluated before multiplication and division.

3

Perform the multiplication and division operations.

-20 + \frac{-36}{-2} - 9 \Rightarrow -20 + 18 - 9

Multiplication and division are processed from left to right before addition and subtraction.

4

Complete the evaluation by performing addition and subtraction from left to right.

-20 + 18 - 9 = -2 - 9 = -11

Addition and subtraction are the final steps in the order of operations.

Anahtar Kavram

Evaluating algebraic expressions requires careful substitution of values using parentheses and strict adherence to the order of operations (PEMDAS).

Soru 5050Soru →

The dimensions of a rectangular box are given by the expressions $5x$ , $(2x - 3)$ , and $(x + 4)$ . The volume of the box is the product of its three dimensions. When the volume is expressed as a fully simplified polynomial in the form $ax^3 + bx^2 + cx$ , determine the value of the coefficient $b$ .

Aşağıdaki boşlukları doldurun

The value of

b

is .

Cevabı ve açıklamayı göster

Cevap

25

To find the coefficient

b

, you must expand the product of the three dimensions. First, multiply the binomials using FOIL:

(2x - 3)(x + 4) = 2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12

. Then, distribute

5x

across the resulting trinomial:

5x(2x^2 + 5x - 12) = 10x^3 + 25x^2 - 60x

. Comparing this to the specified polynomial form

ax^3 + bx^2 + cx

, the term with

x^2

is

25x^2

, making the coefficient

b = 25

.

Adım Adım Çözüm

1

Set up the volume expression by multiplying the three given dimensions.

V = 5x(2x - 3)(x + 4)

The volume of a rectangular box is calculated as length

\times

width

\times

height.

2

Multiply the two binomials using the FOIL method.

(2x - 3)(x + 4) = 2x^2 + 8x - 3x - 12 = 2x^2 + 5x - 12

It is mathematically sound and often easier to multiply the binomials first before distributing the monomial.

3

Distribute the monomial

5x

to each term in the resulting trinomial.

5x(2x^2 + 5x - 12) = 10x^3 + 25x^2 - 60x

To fully expand the polynomial, every term inside the parentheses must be multiplied by

5x

.

4

Identify the value of

b

by matching the result to the given standard form

ax^3 + bx^2 + cx

.

b = 25

The variable

b

represents the coefficient of the

x^2

term in the fully expanded polynomial.

Anahtar Kavram

Multiplying Polynomials and the Distributive Property

Soru 5051Soru →

Given the values $x = \frac{1}{2}$ , $y = -4$ , and $z = -2$ , determine the value of the expression $4x^2y - yz^3$ .

Cevabı ve açıklamayı göster

Cevap: -36

Cevap

-36

The correct answer is derived by substituting the variables accurately, applying the exponents (especially noting that a negative base to an odd power remains negative), and strictly following the order of operations to multiply and then subtract.

Adım Adım Çözüm

1

Substitute

x = \frac{1}{2}

,

y = -4

, and

z = -2

into the expression.

4(\frac{1}{2})^2(-4) - (-4)(-2)^3

Variable substitution must be done exactly as given to prepare for evaluation.

2

Evaluate the terms with exponents.

(\frac{1}{2})^2 = \frac{1}{4}

and

(-2)^3 = -8

. The expression becomes

4(\frac{1}{4})(-4) - (-4)(-8)

.

According to the order of operations (PEMDAS), exponents must be resolved before multiplication.

3

Perform the multiplication operations from left to right.

4(\frac{1}{4})(-4) = 1(-4) = -4

and

(-4)(-8) = 32

. The expression is now

-4 - 32

.

Multiplication is the next step in the order of operations. Note that multiplying two negative numbers results in a positive product.

4

Perform the final subtraction.

-4 - 32 = -36

Subtracting a positive number from a negative number moves further left on the number line.

Anahtar Kavram

Evaluating algebraic expressions using rational numbers, negative bases, and the correct order of operations.

Tahmini Süre:1m 0s

Soru 5052Soru →

If the polynomial expression $(3x - 5)$ is multiplied by the expression $(Ax^2 + 2x - 7)$ , the resulting simplified polynomial has an $x^2$ coefficient of $-19$ . What is the value of the constant $A$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

The

x^2

terms in the product of

(3x - 5)

and

(Ax^2 + 2x - 7)

are generated by multiplying the

x

terms together

(3x \cdot 2x = 6x^2)

and by multiplying the constant by the

x^2

term

(-5 \cdot Ax^2 = -5Ax^2)

. Setting their sum equal to

-19x^2

gives the equation

6 - 5A = -19

. Solving for

A

yields

A = 5

.

Adım Adım Çözüm

1

Identify combinations that yield an

x^2

term

The

x^2

terms come from multiplying

(3x)

by

(2x)

and

(-5)

by

(Ax^2)

.

To find the coefficient of a specific term without expanding the entire polynomial, only multiply the terms whose exponents sum to the target power.

2

Calculate the partial products

(3x)(2x) = 6x^2

and

(-5)(Ax^2) = -5Ax^2

These represent the only contributions to the

x^2

term in the final expanded polynomial.

3

Formulate an equation for the coefficient

6 - 5A = -19

The sum of the partial

x^2

coefficients must equal the total

x^2

coefficient given in the problem.

4

Solve the linear equation for

A

-5A = -25 \rightarrow A = 5

Isolating the variable

A

provides the final answer.

Anahtar Kavram

Multiplying Polynomials

Soru 5053Soru →

Calculate the result of the following algebraic fraction when $x = -2$ and $y = 2$ .

Aşağıdaki boşlukları doldurun

The evaluated value of the expression

\frac{4x^3 + 2y^2}{x - y}

is .

Cevabı ve açıklamayı göster

Cevap

6

The correct answer is 6 because substituting the values

x = -2

and

y = 2

into the expression yields a numerator of

-24

and a denominator of

-4

. Dividing a negative by a negative results in a positive 6.

Adım Adım Çözüm

1

Substitute the given values for

x

and

y

into the expression.

\frac{4(-2)^3 + 2(2)^2}{-2 - 2}

Variable substitution is the first required step in evaluating any algebraic expression.

2

Evaluate the terms with exponents in the numerator.

(-2)^3 = -8

and

2^2 = 4

According to the order of operations (PEMDAS), exponents must be resolved before multiplication.

3

Perform the multiplication operations in the numerator.

4(-8) = -32

and

2(4) = 8

Multiplication is performed after exponents.

4

Add the resulting terms in the numerator and subtract the terms in the denominator.

Numerator:

-32 + 8 = -24

. Denominator:

-2 - 2 = -4

.

Simplify the numerator and denominator completely before dividing.

5

Divide the simplified numerator by the simplified denominator.

\frac{-24}{-4} = 6

Dividing a negative number by a negative number results in a positive quotient.

Anahtar Kavram

Evaluating algebraic expressions with negative integers and exponents

Tahmini Süre:1m 0s

Soru 5054Soru →

A logistics specialist is calculating the weight of four different cargo pallets in kilograms. Based on the provided mathematical expressions for each pallet, arrange them in order from lightest to heaviest.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct order from lightest to heaviest is Pallet S (

47.95

kg), Pallet Q (

48.1

kg), Pallet P (

48.2

kg), and Pallet R (

48.4

kg).

The correct sequence is determined by first applying the proper order of operations (PEMDAS) to compute the exact weight of each pallet. Multiplying and dividing before adding and subtracting yields the values: Pallet S is

47.95

kg, Pallet Q is

48.1

kg, Pallet P is

48.2

kg, and Pallet R is

48.4

kg. Comparing these final decimal figures starting from the greatest place value provides the correct ascending order.

Adım Adım Çözüm

1

Calculate the exact weight of Pallet S.

9.5 \times 5.4 - 3.35 = 51.3 - 3.35 = 47.95

Applying the order of operations, multiplication is performed before subtraction.

2

Calculate the exact weight of Pallet Q.

12.5 + 7.12 \times 5 = 12.5 + 35.6 = 48.1

Multiplication precedes addition. Aligning the decimals properly ensures correct addition.

3

Calculate the exact weight of Pallet P.

60.2 - 4.8 \times 2.5 = 60.2 - 12 = 48.2

Multiply first to get exactly

12

, then subtract from

60.2

.

4

Calculate the exact weight of Pallet R.

58.08 \div 1.2 = 580.8 \div 12 = 48.4

Shift the decimal point one place to the right in both the dividend and divisor to compute the division.

5

Compare the final decimal values to determine the ascending order.

47.95 < 48.1 < 48.2 < 48.4

Comparing digits from left to right (tens, ones, tenths, hundredths) confirms the sequence.

Anahtar Kavram

Applying the order of operations (PEMDAS) to evaluate decimal expressions and comparing relative decimal values.

Soru 5055Soru →

A polynomial $P(x)$ is defined as the product of the binomial $(4x - 3)$ and the trinomial $(-2x^2 + 5x - 1)$ . What is the fully expanded and simplified form of $P(x)$ ?

-8x^3 + 14x^2 - 19x - 3

-2x^2 + x + 3

-8x^3 + 26x^2 - 19x + 3

-8x^3 - 15x + 3

Cevabı ve açıklamayı göster

Cevap: $-8x^3 + 26x^2 - 19x + 3$

Cevap

-8x^3 + 26x^2 - 19x + 3

The correct expression is found by successfully distributing both

4x

and

-3

across the trinomial

(-2x^2 + 5x - 1)

, maintaining correct exponents and signs, and finally combining the like terms:

20x^2

with

6x^2

, and

-4x

with

-15x

.

Adım Adım Çözüm

1

Set up the multiplication using the distributive property.

4x(-2x^2 + 5x - 1) - 3(-2x^2 + 5x - 1)

Every term in the first polynomial must be multiplied by every term in the second polynomial.

2

Distribute

4x

through the trinomial.

4x(-2x^2) + 4x(5x) + 4x(-1) = -8x^3 + 20x^2 - 4x

Multiply coefficients and add exponents for the variable base.

3

Distribute

-3

through the trinomial.

-3(-2x^2) - 3(5x) - 3(-1) = 6x^2 - 15x + 3

Multiply the constant by each term, carefully applying integer sign rules.

4

Combine the results and group like terms.

-8x^3 + (20x^2 + 6x^2) + (-4x - 15x) + 3

Grouping terms with the same variable exponent prepares the expression for final simplification.

5

Add the coefficients of the like terms.

-8x^3 + 26x^2 - 19x + 3

This yields the polynomial in standard form.

Anahtar Kavram

Multiplying polynomials by distributing every term in the first factor to every term in the second, then combining like terms.

Soru 5056Soru →

Evaluate the given algebraic equation to solve for $p$ :

7 - 3(p + 2) = p - 7

Which of the following represents the correct solution for $p$ ?

5

-5

-4

2

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The correct value is 2.

The correct approach involves distributing the

-3

into the parentheses to get

7 - 3p - 6

, combining like terms to get

1 - 3p

, and then balancing the equation to solve for

p

. Adding

3p

to both sides yields

1 = 4p - 7

, and adding

7

gives

8 = 4p

, which simplifies to

p = 2

.

Adım Adım Çözüm

1

Apply the distributive property to the left side of the equation.

7 - 3p - 6 = p - 7

The -3 must be multiplied by both

p

and

2

to eliminate the parentheses.

2

Combine the constant terms on the left side.

1 - 3p = p - 7

Combining

7

and

-6

simplifies the equation before moving terms across the equals sign.

3

Add

3p

to both sides of the equation.

1 = 4p - 7

This groups all terms containing the variable

p

on the right side.

4

Add

7

to both sides and divide by

4

.

8 = 4p \Rightarrow p = 2

This isolates the variable

p

to find the final numerical solution.

Anahtar Kavram

Solving Linear Equations with Distribution

Soru 5057Soru →

A military field generator consumes diesel fuel at a rate of $3.25$ gallons per hour. The generator's fuel tank initially contains $50.4$ gallons of fuel. If the generator runs continuously for $8.4$ hours, how many gallons of fuel remain in the tank?

Cevabı ve açıklamayı göster

Cevap: 23.1

Cevap

23.1

To find the remaining fuel, you first need to determine the total amount used. Multiplying the hourly rate of

3.25

gallons by

8.4

hours gives

27.3

gallons consumed. Subtracting this from the initial

50.4

gallons (

50.4 - 27.3

) leaves exactly

23.1

gallons.

Adım Adım Çözüm

1

Calculate the total amount of fuel consumed by the generator during the run time.

3.25 \times 8.4 = 27.3

gallons

The total fuel consumed is found by multiplying the hourly consumption rate by the total number of hours the generator operates.

2

Calculate the remaining fuel by subtracting the consumed fuel from the initial amount.

50.4 - 27.3 = 23.1

gallons

The fuel left in the tank is the difference between the starting volume and the volume that was burned.

Anahtar Kavram

Solving multi-step practical arithmetic problems involving decimal multiplication and decimal subtraction.

Tahmini Süre:1m 0s

Soru 5058Soru →

A manufacturer is cutting a square opening out of a rectangular metal panel. The rectangular panel has a length of $(4x - 1)$ inches and a width of $(2x + 3)$ inches. The square opening to be removed has a side length of $(x + 2)$ inches. Which of the following expressions represents the total remaining area of the metal panel after the square is removed?

7x^2 + 6x - 7

7x^2 + 10x - 7

7x^2 + 14x + 1

7x^2 - 7

Cevabı ve açıklamayı göster

Cevap: $7x^2 + 6x - 7$

Cevap

The correct expression is

7x^2 + 6x - 7

.

The remaining area is found by subtracting the square's area from the rectangle's area. Multiplying the dimensions

(4x - 1)(2x + 3)

yields

8x^2 + 10x - 3

. Squaring the dimension

(x + 2)

yields

x^2 + 4x + 4

. Subtracting the entire second polynomial requires distributing the negative sign, resulting in

8x^2 + 10x - 3 - x^2 - 4x - 4

, which correctly simplifies to

7x^2 + 6x - 7

.

Adım Adım Çözüm

1

Calculate the area of the rectangular panel.

(4x - 1)(2x + 3) = 8x^2 + 12x - 2x - 3 = 8x^2 + 10x - 3

The area of a rectangle is length multiplied by width, which requires multiplying two binomials using the FOIL method.

2

Calculate the area of the square opening.

(x + 2)^2 = (x + 2)(x + 2) = x^2 + 2x + 2x + 4 = x^2 + 4x + 4

The area of a square is its side length multiplied by itself.

3

Subtract the square's area from the rectangle's area.

(8x^2 + 10x - 3) - (x^2 + 4x + 4) = 8x^2 + 10x - 3 - x^2 - 4x - 4 = 7x^2 + 6x - 7

To find the remaining area, the removed portion must be subtracted. It is critical to distribute the negative sign to every term inside the parentheses being subtracted.

Anahtar Kavram

Multiplying binomials and subtracting polynomials while properly distributing negative signs.

Soru 5059Soru →

Which of the following represents the value of $y$ that makes the given equation true?

\frac{2y - 5}{3} + 8 = 3

-5

0

5

19

Cevabı ve açıklamayı göster

Cevap: -5

Cevap

The correct value for the variable is -5.

The solution requires a sequence of inverse operations to isolate the variable. First, subtracting 8 from both sides gives a value of -5 on the right. Then, multiplying by 3 clears the fraction, resulting in 2y - 5 = -15. Adding 5 yields 2y = -10, and dividing by 2 confirms the final value is -5.

Adım Adım Çözüm

1

Isolate the fractional term by subtracting 8 from both sides of the equation.

\frac{2y - 5}{3} = -5

Applying the inverse operation of addition moves the constant term to the right side, simplifying the equation.

2

Multiply both sides of the equation by 3.

2y - 5 = -15

Multiplying by the denominator eliminates the fraction, making it possible to solve for the variable.

3

Add 5 to both sides of the equation.

2y = -10

This step isolates the term with the variable on the left side.

4

Divide both sides by 2.

y = -5

Dividing by the coefficient of the variable gives the final exact value.

Anahtar Kavram

Solving one-variable linear equations involving fractions

Soru 5060Soru →

A student is asked to simplify the expression below.

8.4 - 2.1 \times (-3.2) + 1.5

What is the correct simplified value?

-18.66

3.18

7.71

16.62

Cevabı ve açıklamayı göster

Cevap: 16.62

Cevap

The correct simplified value is 16.62.

The exact evaluation requires multiplying 2.1 by -3.2 first to obtain -6.72. The expression then becomes an addition problem (

8.4 + 6.72 + 1.5

). Properly aligning the decimal points and summing these terms yields the true simplified value of 16.62.

Adım Adım Çözüm

1

Identify and perform the multiplication operation first.

2.1 \times (-3.2) = -6.72

Following the order of operations (PEMDAS), multiplication takes precedence over addition and subtraction.

2

Substitute the product back into the expression and resolve the subtraction of a negative value.

8.4 - (-6.72) + 1.5 = 8.4 + 6.72 + 1.5

Subtracting a negative number is mathematically identical to adding its positive counterpart.

3

Perform the addition operations from left to right, making sure to properly align the decimal points.

8.40 + 6.72 + 1.50 = 16.62

Addition operations are performed sequentially from left to right once all higher-priority operations are resolved, requiring proper place-value alignment.

Anahtar Kavram

Decimal Arithmetic and Order of Operations

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