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Decimal Operations

16 questions

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Question 1Question →

A drone is flying at an altitude of 45.545.5 meters. It descends 12.7512.75 meters to take a photo, and then ascends 8.28.2 meters. What is the new altitude of the drone?

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Answer: 40.95 meters

Answer

The new altitude of the drone is 40.95 meters.
The correct answer is found by sequentially subtracting the descent from the initial altitude (45.5−12.75=32.7545.5 - 12.75 = 32.75) and then adding the ascent (32.75+8.2=40.9532.75 + 8.2 = 40.95). Aligning the decimal points correctly ensures an accurate calculation.

Step-by-Step Solution

1
Identify the starting altitude and the mathematical operations for descending and ascending.
Starting altitude is 45.545.5 meters. Descending means subtracting (12.7512.75), and ascending means adding (8.28.2).
Translating the real-world scenario into a mathematical expression ensures the correct operations are applied in the correct sequence.
2
Subtract the descent distance from the initial altitude.
45.5−12.75=32.7545.5 - 12.75 = 32.75
This finds the drone's intermediate altitude after its initial descent.
3
Add the ascent distance to the intermediate altitude, making sure to align the decimal points.
32.75+8.20=40.9532.75 + 8.20 = 40.95
This calculates the final altitude after the drone moves back up.

Key Concept

Decimal Operations
Question 2Question →

Which of the following represents the correct evaluation of the mathematical expression below?

−1.5−2.4×(6.2−3.45)-1.5 - 2.4 \times (6.2 - 3.45)
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Answer: -8.1

Answer

The correct answer is -8.1.
The correct evaluation requires following the exact order of operations. First, correctly align the decimals to subtract inside the parentheses (6.20−3.45=2.756.20 - 3.45 = 2.75). Second, multiply that result by 2.42.4 to get 6.66.6. Finally, calculate −1.5−6.6-1.5 - 6.6, which correctly yields −8.1-8.1.

Step-by-Step Solution

1
Evaluate the expression inside the parentheses.
6.20−3.45=2.756.20 - 3.45 = 2.75
According to the order of operations (PEMDAS), parentheses must be resolved first. A zero is added to 6.26.2 to properly align the decimals.
2
Perform the multiplication.
−2.4×2.75=−6.6-2.4 \times 2.75 = -6.6
Multiplication takes precedence over addition and subtraction. The product of a negative and a positive number is negative.
3
Perform the final subtraction.
−1.5−6.6=−8.1-1.5 - 6.6 = -8.1
Combine the remaining terms. Subtracting a positive number from a negative number moves further left on the number line.

Key Concept

Order of Operations with Decimals
Question 3Question →

Arrange the following mathematical expressions in ascending order, from the least value to the greatest value.

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Answer

The correct order from least to greatest is: 1.2×0.61.2 \times 0.6, then 0.35+0.380.35 + 0.38, then 4.2−3.454.2 - 3.45, and finally 0.48÷0.60.48 \div 0.6.
Each arithmetic expression must be evaluated independently applying the correct decimal rules for that specific operation. The results are 0.720.72, 0.730.73, 0.750.75, and 0.800.80. Sorting these decimal values chronologically by magnitude yields the correct ascending sequence.

Step-by-Step Solution

1
Evaluate the multiplication expression: 1.2×0.61.2 \times 0.6
0.720.72
Multiply the whole numbers (12×6=7212 \times 6 = 72) and count a total of two decimal places from the factors.
2
Evaluate the addition expression: 0.35+0.380.35 + 0.38
0.730.73
Align the decimal points vertically and add the hundredths, then the tenths.
3
Evaluate the subtraction expression: 4.2−3.454.2 - 3.45
0.750.75
Rewrite 4.24.2 as 4.204.20 to align the decimal points properly, then subtract.
4
Evaluate the division expression: 0.48÷0.60.48 \div 0.6
0.800.80
Multiply both numbers by 1010 to make the divisor a whole number (4.8÷64.8 \div 6), then divide.
5
Order the evaluated results from least to greatest.
0.72<0.73<0.75<0.800.72 < 0.73 < 0.75 < 0.80
Comparing the values digit by digit from the tenths place to the hundredths place establishes the correct magnitude sequence.

Key Concept

Evaluating all four basic operations with decimals and comparing their relative magnitudes.
Question 4Question →

What is the value of the following expression?

8.4−2.4×1.5+4.5÷0.58.4 - 2.4 \times 1.5 + 4.5 \div 0.5
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Answer: 13.8

Answer

13.8
Applying the order of operations correctly requires first evaluating the multiplication (2.4×1.5=3.62.4 \times 1.5 = 3.6) and the division (4.5÷0.5=94.5 \div 0.5 = 9). Substituting these values back simplifies the equation to 8.4−3.6+98.4 - 3.6 + 9. Processing the remaining subtraction and addition from left to right yields 4.8+9=13.84.8 + 9 = 13.8.

Step-by-Step Solution

1
Evaluate the multiplication part of the expression.
2.4×1.5=3.62.4 \times 1.5 = 3.6
According to the order of operations, multiplication must be performed before addition or subtraction.
2
Evaluate the division part of the expression.
4.5÷0.5=94.5 \div 0.5 = 9
Division shares the same precedence as multiplication and must also be evaluated before addition and subtraction.
3
Substitute the results back into the original expression.
8.4−3.6+98.4 - 3.6 + 9
This simplifies the expression so that only addition and subtraction remain.
4
Evaluate the remaining addition and subtraction from left to right.
8.4−3.6=4.88.4 - 3.6 = 4.8, and 4.8+9=13.84.8 + 9 = 13.8
Addition and subtraction have equal precedence and are solved strictly from left to right.

Key Concept

Order of Operations with Decimals
Question 5Question →

Evaluate the following mathematical expressions involving decimals. What is the correct sequence of these expressions when ordered from least to greatest value?

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Answer

The correct order from least to greatest is: 0.15×40−14.4÷1.20.15 \times 40 - 14.4 \div 1.2, followed by 10.5−3.2×2.510.5 - 3.2 \times 2.5, then (4.2−1.8)÷0.6+0.5(4.2 - 1.8) \div 0.6 + 0.5, and finally 2.4×0.5+1.8÷0.32.4 \times 0.5 + 1.8 \div 0.3.
Evaluating each expression using the correct order of operations yields final decimal values of 7.27.2, 2.52.5, −6.0-6.0, and 4.54.5. Arranging these values numerically from least to greatest provides the correct structural sequence.

Step-by-Step Solution

1
Evaluate the first expression: 2.4×0.5+1.8÷0.32.4 \times 0.5 + 1.8 \div 0.3.
1.2+6=7.21.2 + 6 = 7.2
According to the order of operations, multiplication and division are performed left-to-right before addition.
2
Evaluate the second expression: 10.5−3.2×2.510.5 - 3.2 \times 2.5.
10.5−8.0=2.510.5 - 8.0 = 2.5
Multiplication must be performed before subtraction.
3
Evaluate the third expression: 0.15×40−14.4÷1.20.15 \times 40 - 14.4 \div 1.2.
6.0−12=−6.06.0 - 12 = -6.0
Perform multiplication and division first. Then subtract the larger value from the smaller value, resulting in a negative number.
4
Evaluate the fourth expression: (4.2−1.8)÷0.6+0.5(4.2 - 1.8) \div 0.6 + 0.5.
2.4÷0.6+0.5=4+0.5=4.52.4 \div 0.6 + 0.5 = 4 + 0.5 = 4.5
Operations inside parentheses take highest priority, followed by division, and finally addition.
5
Order the evaluated results from least to greatest.
−6.0<2.5<4.5<7.2-6.0 < 2.5 < 4.5 < 7.2
Arranging the numerical values based on their algebraic magnitude provides the final sequence.

Key Concept

Order of Operations with Decimals

Alternative Method

Convert all terminating decimals to fractions (e.g., 0.5=120.5 = \frac{1}{2}, 0.3=3100.3 = \frac{3}{10}) before performing the operations to avoid decimal placement and alignment errors.
Estimated Time:1m 30s
Question 6Question →

Calculate the value of each decimal expression below. Arrange the results in ascending order (from smallest to largest).

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Answer

The correct order from smallest to largest is: 1.5×21.5 \times 2, followed by 1.2+2.41.2 + 2.4, then 8.4÷28.4 \div 2, and finally 5.8−1.35.8 - 1.3.
Evaluating each mathematical expression individually yields 4.5, 3.6, 4.2, and 3.0 respectively. Arranging these numerical results in ascending order gives 3.0, 3.6, 4.2, and 4.5. This directly maps to the expressions: 1.5×21.5 \times 2, then 1.2+2.41.2 + 2.4, then 8.4÷28.4 \div 2, and finally 5.8−1.35.8 - 1.3.

Step-by-Step Solution

1
Calculate the value of the first expression.
5.8−1.3=4.55.8 - 1.3 = 4.5
To determine the numerical value for comparison.
2
Calculate the value of the second expression.
1.2+2.4=3.61.2 + 2.4 = 3.6
To determine the numerical value for comparison.
3
Calculate the value of the third expression.
8.4÷2=4.28.4 \div 2 = 4.2
To determine the numerical value for comparison.
4
Calculate the value of the fourth expression.
1.5×2=3.01.5 \times 2 = 3.0
To determine the numerical value for comparison.
5
Compare the four calculated values and order them from smallest to largest.
3.0<3.6<4.2<4.53.0 < 3.6 < 4.2 < 4.5
To establish the final ascending sequence required by the question.

Key Concept

Performing basic operations (addition, subtraction, multiplication, division) with decimals and comparing their magnitudes.
Estimated Time:45s
Question 7Question →

A plumber has a copper pipe that is 14.514.5 inches long. She cuts off a piece that is 3.753.75 inches long. What is the length of the remaining pipe in inches?

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Answer: 10.75

Answer

10.75
Subtracting the cut piece (3.753.75 inches) from the total length (14.514.5 inches) gives the remaining length. Aligning the decimals properly as 14.50−3.7514.50 - 3.75 results in exactly 10.7510.75.

Step-by-Step Solution

1
Identify the operation needed
Subtraction: 14.5−3.7514.5 - 3.75
Cutting a piece off means reducing the total length of the pipe.
2
Align the decimals for subtraction
14.50−3.7514.50 - 3.75
To subtract decimals correctly, align the place values by lining up the decimal points. Add a placeholder zero to 14.514.5 to make it 14.5014.50.
3
Perform the subtraction
10.7510.75
Subtracting 3.753.75 from 14.5014.50 yields 10.7510.75.

Key Concept

Decimal Subtraction and Alignment
Estimated Time:45s
Question 8Question →

Determine the final value of each arithmetic expression below. What is the correct sequence when the expressions are arranged from smallest to largest result?

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Answer

The correct sequence from smallest to largest is: 0.18+0.420.18 + 0.42, 3.5÷53.5 \div 5, 2.1−1.32.1 - 1.3, and 0.15×60.15 \times 6.
To arrange the sequence properly, first calculate their values: 0.18+0.42=0.60.18 + 0.42 = 0.6, 3.5÷5=0.73.5 \div 5 = 0.7, 2.1−1.3=0.82.1 - 1.3 = 0.8, and 0.15×6=0.90.15 \times 6 = 0.9. By comparing these results side by side, the accurate ascending order is 0.60.6, 0.70.7, 0.80.8, and 0.90.9.

Step-by-Step Solution

1
Calculate the sum for the fourth expression.
0.18+0.42=0.600.18 + 0.42 = 0.60, which is 0.60.6.
Adding the hundredths and tenths places properly yields 0.60.6.
2
Calculate the quotient for the third expression.
3.5÷5=0.73.5 \div 5 = 0.7.
Since 35÷5=735 \div 5 = 7, dividing 3.53.5 by 55 gives 0.70.7.
3
Calculate the difference for the first expression.
2.1−1.3=0.82.1 - 1.3 = 0.8.
Borrowing from the ones place allows subtracting 1.31.3 from 2.12.1 to get 0.80.8.
4
Calculate the product for the second expression.
0.15×6=0.900.15 \times 6 = 0.90, which is 0.90.9.
Multiplying 1515 by 66 gives 9090. Since the original factors have a total of two decimal places, the result is 0.900.90.
5
Order the evaluated results from smallest to largest.
0.6<0.7<0.8<0.90.6 < 0.7 < 0.8 < 0.9.
Comparing the tenths digit of each calculated value provides the final arrangement.

Key Concept

Performing basic arithmetic operations with decimals and comparing their magnitudes.
Question 9Question →

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

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Answer: −8.52-8.52

Answer

The correct value is −8.52-8.52.
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=−1.21.5 - 2.7 = -1.2. Next, the multiplication is performed before the addition: 3.6×(−1.2)=−4.323.6 \times (-1.2) = -4.32. Finally, −4.32-4.32 is added to −4.2-4.2, making sure to properly align the decimal points, which yields −8.52-8.52.

Step-by-Step Solution

1
Evaluate the expression inside the parentheses.
1.5−2.7=−1.21.5 - 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 3.63.6.
3.6×(−1.2)=−4.323.6 \times (-1.2) = -4.32
Multiplication takes precedence over addition.
3
Add −4.2-4.2 to the product from step 2.
−4.2+(−4.32)=−8.52-4.2 + (-4.32) = -8.52
This is the final remaining operation. Decimal points must be aligned vertically (−4.20+(−4.32)-4.20 + (-4.32)) to ensure correct addition.

Key Concept

Decimal Operations with Order of Operations
Question 10Question →

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.

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Answer

The correct order from lightest to heaviest is Pallet S (47.9547.95 kg), Pallet Q (48.148.1 kg), Pallet P (48.248.2 kg), and Pallet R (48.448.4 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 47.9547.95 kg, Pallet Q is 48.148.1 kg, Pallet P is 48.248.2 kg, and Pallet R is 48.448.4 kg. Comparing these final decimal figures starting from the greatest place value provides the correct ascending order.

Step-by-Step Solution

1
Calculate the exact weight of Pallet S.
9.5×5.4−3.35=51.3−3.35=47.959.5 \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+7.12×5=12.5+35.6=48.112.5 + 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.
60.2−4.8×2.5=60.2−12=48.260.2 - 4.8 \times 2.5 = 60.2 - 12 = 48.2
Multiply first to get exactly 1212, then subtract from 60.260.2.
4
Calculate the exact weight of Pallet R.
58.08÷1.2=580.8÷12=48.458.08 \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<48.447.95 < 48.1 < 48.2 < 48.4
Comparing digits from left to right (tens, ones, tenths, hundredths) confirms the sequence.

Key Concept

Applying the order of operations (PEMDAS) to evaluate decimal expressions and comparing relative decimal values.
Question 11Question →

A military field generator consumes diesel fuel at a rate of 3.253.25 gallons per hour. The generator's fuel tank initially contains 50.450.4 gallons of fuel. If the generator runs continuously for 8.48.4 hours, how many gallons of fuel remain in the tank?

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Answer: 23.1

Answer

23.1
To find the remaining fuel, you first need to determine the total amount used. Multiplying the hourly rate of 3.253.25 gallons by 8.48.4 hours gives 27.327.3 gallons consumed. Subtracting this from the initial 50.450.4 gallons (50.4−27.350.4 - 27.3) leaves exactly 23.123.1 gallons.

Step-by-Step Solution

1
Calculate the total amount of fuel consumed by the generator during the run time.
3.25×8.4=27.33.25 \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=23.150.4 - 27.3 = 23.1 gallons
The fuel left in the tank is the difference between the starting volume and the volume that was burned.

Key Concept

Solving multi-step practical arithmetic problems involving decimal multiplication and decimal subtraction.
Estimated Time:1m 0s
Question 12Question →

A student is asked to simplify the expression below.

8.4−2.1×(−3.2)+1.58.4 - 2.1 \times (-3.2) + 1.5

What is the correct simplified value?

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Answer: 16.62

Answer

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+1.58.4 + 6.72 + 1.5). Properly aligning the decimal points and summing these terms yields the true simplified value of 16.62.

Step-by-Step Solution

1
Identify and perform the multiplication operation first.
2.1×(−3.2)=−6.722.1 \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+1.58.4 - (-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=16.628.40 + 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.

Key Concept

Decimal Arithmetic and Order of Operations
Question 13Question →

What is the value of the following mathematical expression?

0.840.04+2.5×(−1.4)\frac{0.84}{0.04} + 2.5 \times (-1.4)
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Answer: 17.5

Answer

17.5
To evaluate the expression correctly, the order of operations must be followed. First, resolve the division: 0.84÷0.04=210.84 \div 0.04 = 21. Next, resolve the multiplication: 2.5×(−1.4)=−3.52.5 \times (-1.4) = -3.5. Finally, add these two results together: 21+(−3.5)=17.521 + (-3.5) = 17.5. Proper decimal alignment (21.0−3.521.0 - 3.5) ensures the correct final value.

Step-by-Step Solution

1
Perform the division operation.
0.840.04=844=21\frac{0.84}{0.04} = \frac{84}{4} = 21
According to the order of operations (PEMDAS), division and multiplication must be evaluated before addition.
2
Perform the multiplication operation.
2.5×(−1.4)=−3.52.5 \times (-1.4) = -3.5
Complete the remaining multiplication before moving on to addition. Note that a positive multiplied by a negative results in a negative product.
3
Add the results of the division and multiplication together.
21+(−3.5)=21.0−3.5=17.521 + (-3.5) = 21.0 - 3.5 = 17.5
Substitute the evaluated terms back into the expression and align the decimal points to subtract correctly.

Key Concept

Order of Operations with Decimals
Estimated Time:1m 0s
Question 14Question →

An industrial pump dispenses 3.453.45 liters of a chemical per cycle. A tank initially contains 150.8150.8 liters of the chemical. If the pump runs for 1414 cycles to remove the chemical from the tank, and then a technician manually adds 5.255.25 liters back into the tank, how many liters of chemical remain in the tank?

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Answer: 107.75

Answer

107.75
The final volume is accurately found by multiplying the removal rate by the number of cycles to determine the total removed amount, subtracting that from the initial volume, and finally adding the manually returned amount while maintaining proper decimal alignment.

Step-by-Step Solution

1
Calculate the total volume of chemical removed by the pump.
14×3.45=48.314 \times 3.45 = 48.3
The pump runs for 14 cycles, removing 3.45 liters each time, so multiplication is required to find the total removal.
2
Subtract the removed volume from the initial tank volume.
150.8−48.3=102.5150.8 - 48.3 = 102.5
Removing the chemical decreases the total volume available in the tank.
3
Add the volume of the chemical manually poured back into the tank.
102.5+5.25=107.75102.5 + 5.25 = 107.75
Adding chemical increases the total volume to its final amount.

Key Concept

Solving multi-step practical problems involving the multiplication, subtraction, and addition of decimals.
Estimated Time:1m 0s
Question 15Question →

What is the final result of the expression 9.25−3.4×0.59.25 - 3.4 \times 0.5?

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Answer: 7.557.55

Answer

The correct answer is 7.557.55.
The expression requires following the proper order of operations, meaning the multiplication must be calculated before the subtraction. First, multiplying 3.43.4 by 0.50.5 yields 1.71.7. Then, aligning the decimals correctly and subtracting 1.71.7 from 9.259.25 provides the final answer of 7.557.55.

Step-by-Step Solution

1
Identify the correct order of operations (PEMDAS/BODMAS) for the expression.
Multiplication must be performed before subtraction.
Standard mathematical rules require evaluating multiplication before addition or subtraction.
2
Multiply 3.43.4 by 0.50.5.
3.4×0.5=1.73.4 \times 0.5 = 1.7
Resolving the multiplication part of the expression.
3
Subtract the product from 9.259.25, ensuring the decimal points are vertically aligned.
9.25−1.70=7.559.25 - 1.70 = 7.55
Completing the final subtraction operation to find the total value.

Key Concept

Decimal Operations
Estimated Time:45s
Question 16Question →

An engineer uses the formula V=V0−r×t+CV = V_0 - r \times t + C to calculate the final volume of fluid in a tank in liters. If the initial volume V0=12.4V_0 = 12.4, the rate r=2.5r = 2.5, the time t=3.2t = 3.2, and a correction factor C=4.8C = 4.8, what is the final volume VV?

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Answer: 9.2

Answer

9.2
To find the final volume, substitute the values into the formula to get 12.4−2.5×3.2+4.812.4 - 2.5 \times 3.2 + 4.8. According to the order of operations, the multiplication must be evaluated first: 2.5×3.2=8.02.5 \times 3.2 = 8.0. Substituting this back gives 12.4−8.0+4.812.4 - 8.0 + 4.8. Performing the remaining operations from left to right, first subtract (12.4−8.0=4.412.4 - 8.0 = 4.4), and then add (4.4+4.8=9.24.4 + 4.8 = 9.2).

Step-by-Step Solution

1
Substitute the given values into the formula.
V=12.4−2.5×3.2+4.8V = 12.4 - 2.5 \times 3.2 + 4.8
This sets up the arithmetic expression that needs to be evaluated.
2
Apply the order of operations (PEMDAS) by performing the multiplication first.
2.5×3.2=8.02.5 \times 3.2 = 8.0
Multiplication takes precedence over addition and subtraction.
3
Rewrite the expression with the product and perform the subtraction (working left to right).
12.4−8.0=4.412.4 - 8.0 = 4.4
Addition and subtraction are evaluated from left to right in the order they appear.
4
Perform the final addition step.
4.4+4.8=9.24.4 + 4.8 = 9.2
This yields the final evaluated volume.

Key Concept

Applying the order of operations (PEMDAS) to expressions containing decimals, ensuring proper place-value alignment during addition and subtraction.