1.1 Arithmetic

Arithmetical terms and signs, methods of multiplication and division, fractions and decimals, factors and multiples, weights, measures and conversion factors, ratio and proportion, averages and percentages, areas and volumes, squares, cubes, square and cube roots.

1.1 Arithmetic

1. Addition of Whole Numbers

Addition is the process of combining two or more numbers to get a total or sum. In aviation maintenance, addition helps track inventory, calculate total quantities of parts, or measure cumulative work hours.

Example:

If an engineer is doing a regular check and installs 3 landing gear bolts on one side of the aircraft and 5 on the other side, the total bolts installed would be:

So, 8 landing gear bolts were installed in total.

2. Subtraction of Whole Numbers

Subtraction is the process of removing one quantity from another. In maintenance, it can help track usage or remaining parts.

Example:

An engineer has 20 screws to work with for securing panels. After using 12 screws on one panel, the remaining screws are calculated as follows:

So, 8 screws are left for other tasks.

3. Multiplication of Whole Numbers

Multiplication is used to find the total when there are multiple groups of the same number. This is helpful in calculating repetitive installations or total part requirements.

Example:

Suppose an aircraft requires 4 fuel nozzles per engine, and the aircraft has 2 engines. To find the total number of nozzles needed:

Therefore, 8 fuel nozzles are required for the entire aircraft.

4. Division of Whole Numbers

Division is used to split a quantity into equal parts or determine how many times one number fits into another. In maintenance, it might apply to distributing resources evenly among tasks.

Example:

An engineer has 40 rivets to secure panels on both wings. If the rivets need to be evenly divided, the number of rivets per wing would be:

Each wing gets 20 rivets.

5. Factors and Multiples

A factor of a number divides that number without leaving a remainder. A multiple of a number is the product of that number and an integer. In aviation, factors and multiples can help determine uniform distribution of parts or work.

Example:

Suppose an engineer has 12 bolts and needs to divide them equally into packages for different aircraft sections. The factors of 12 are 1, 2, 3, 4, 6, and 12. The engineer can divide the bolts into any of these quantities to ensure each package has an even distribution.

If bolts need to be organized in packages of 3, we use multiples of 3 (3, 6, 9, 12, etc.) to determine the number of bolts in each package and how many packages can be prepared.

6. Lowest Common Multiple (LCM)

The LCM of two numbers is the smallest multiple shared by both numbers. In aviation, LCM can be used to schedule recurring maintenance tasks that occur on different intervals.

Example:

If Engine A needs an inspection every 6 days and Engine B every 8 days, the LCM of 6 and 8 helps determine when both inspections will align on the same day.

Finding the LCM of 6 and 8: Multiples of 6: 6, 12, 18, 24, … Multiples of 8: 8, 16, 24, …

The smallest common multiple is 24 days, so both inspections will align every 24 days.

7. Highest Common Factor (HCF)

The HCF of two numbers is the largest factor they share. In maintenance, HCF can help in efficiently grouping or sharing parts when limited resources are available.

Example:

An engineer has 36 screws and 60 washers. To create uniform kits with the same number of screws and washers in each, the HCF will determine the maximum number of kits that can be created.

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

The HCF is 12, so the engineer can create 12 kits, each containing 3 screws and 5 washers.

8. Prime Numbers

Prime numbers are numbers greater than 1 that only have two factors: 1 and themselves. Prime numbers are useful when planning inspections that only align on unique time intervals.

Example:

Suppose an engineer tracks inspections that occur on cycles based on prime numbers to avoid overlap with other tasks. Cycles might be set for 3, 5, 7, 11 days, etc., as these numbers won’t coincide with each other as frequently, allowing for a staggered maintenance schedule.

9. Prime Factors

Prime factors are the prime numbers that multiply together to give the original number. Prime factorization can help engineers simplify large numbers for use in formulas or scheduling.

Example:

If an engineer needs to determine how many maintenance hours to allocate over 28 days, the prime factorization of 28 is: This breakdown helps distribute workload evenly based on smaller units (2-day or 7-day cycles).

6. Lowest Common Multiple using Prime Factors Method

This method uses the prime factors of two numbers to find their LCM. This is useful when planning tasks with different schedules, ensuring that overlap occurs at the right intervals.

Example:

For example, if one inspection happens every 12 days and another every 15 days, we find the LCM by using the prime factorization:

Taking the highest power of each prime factor:

So, both tasks will align every 60 days.

7. Highest Common Factor using Prime Factors Method

To find the HCF using prime factors, we identify the lowest powers of common prime factors. This can optimize resource allocation.

Example:

If there are two part quantities, 48 rivets and 60 bolts, and we want to create identical kits with the same quantity of both, we use the HCF.

Prime factorizations:

The lowest powers of the common prime factors are

This allows for 12 kits, each with an equal number of rivets and bolts.

8. Precedence

Precedence, or order of operations, determines the sequence in which arithmetic operations are performed. In aviation calculations, following the correct order is essential to ensure accurate results, especially in complex formulas.

Example:

If an engineer calculates fuel requirements using the formula: According to the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), multiplication is done first:

So, the correct answer is 18 units of fuel.

9. Use of Variables

Variables are symbols used to represent numbers in equations and formulas, making it easier to generalize calculations. In aviation, variables help create formulas for weight distribution, fuel consumption, or other repetitive tasks.

Example:

Let F represent fuel flow in liters per hour, and T represent time in hours. If an engineer needs to calculate total fuel consumption, the formula is $Total Fuel = F \times T$

If liters/hour and hours:

So, 600 liters of fuel are needed.

10. Reciprocal

The reciprocal of a number is 1 divided by that number. Reciprocal values can be used in calculations like finding rates or time inversely.

Example:

An engineer has a rate of 4 parts assembled per hour. The reciprocal of 4 (or gives the time per part:

Thus, each part takes 0.25 hours (15 minutes) to assemble.

11. Positive and Negative Numbers (Signed Numbers)

Positive numbers represent values above zero, while negative numbers represent values below zero. In aviation, signed numbers are crucial for altitude adjustments, temperature changes, and pressure measurements.

Example:

An aircraft is cruising at an altitude of 10,000 feet and descends to 7,000 feet. To find the change in altitude:

Starting altitude = 10,000 feet
New altitude = 7,000 feet

The change in altitude is:

This -3,000 feet represents a descent of 3,000 feet, indicating that the aircraft has moved 3,000 feet downward from its original altitude.

12. Addition of Positive and Negative Numbers

When adding signed numbers, if they have the same sign, add the absolute values and keep the sign; if different, subtract the smaller from the larger and use the sign of the larger.

Example:

Suppose an engineer records temperature changes of +5°C and -3°C in a single day:

The net change is +2°C, indicating a 2-degree increase.

13. Subtraction of Positive and Negative Numbers

When subtracting signed numbers, change the sign of the number being subtracted and proceed with addition.

Example:

If a maintenance log shows a pressure of -4 psi and it decreases by another -6 psi, the calculation is:

So, the pressure change results in +2 psi.

14. Multiplication of Positive and Negative Numbers

In multiplication, if both numbers have the same sign, the product is positive; if different signs, the product is negative.

Example:

If an engineer calculates the effect of a pressure drop (-4) over 3 cycles:

The result is -12 psi total pressure drop.

15. Division of Positive and Negative Numbers

For division, similar to multiplication, if both numbers have the same sign, the result is positive; if different, the result is negative.

Example:

If a load of -15 kg is distributed over 3 supports:

Each support would bear -5 kg, indicating a reduction in load by 5 kg per support.

16. Fractions

A fraction represents a part of a whole and is written as two numbers separated by a line: the numerator (top number) and the denominator (bottom number). Fractions are common in aviation for measurements and quantities.

Example:

If an engineer uses 3/4 of a liter of hydraulic fluid from a container, this fraction shows that 3 parts out of 4 have been used.

17. Finding the Least Common Denominator (LCD)

The least common denominator of two or more fractions is the smallest number that each denominator divides into evenly. LCD is essential when adding or subtracting fractions with different denominators.

Example:

An engineer has two fractions of fuel remaining in two tanks: 1/3 and 1/4. To combine these, find the LCD of 3 and 4.

Multiples of 3: 3, 6, 9, 12, …
Multiples of 4: 4, 8, 12, …

The LCD is 12, so both fractions are converted to have 12 as the denominator:

18. Reducing Fractions

Reducing (or simplifying) a fraction means dividing the numerator and denominator by their greatest common factor (GCF) until the fraction is in its simplest form.

Example:

An engineer measures a bolt at 8/16 inches. Dividing both numerator and denominator by

So, the measurement simplifies to 1/2 inch.

19. Mixed Numbers

A mixed number combines a whole number and a fraction. Mixed numbers are useful in aviation when measurements fall between whole units.

Example:

If a wire is measured at 2 1/2 feet, it means the wire is 2 feet and an additional half foot long.

20. Addition and Subtraction of Fractions

To add or subtract fractions, they must have the same denominator. If they don’t, convert to a common denominator before performing the operation.

Example (Addition):

Two fuel containers have 3/8 and 1/4 liters remaining. Converting 1/4 to 2/8 (common denominator 8):

So, 5/8 liter of fuel remains in total.

Example (Subtraction):

If 3/4 liter of oil is used and an additional 1/4 liter is needed, the remaining amount is:

So, 1/2 liter of oil is still available.

21. Multiplication of Fractions

When multiplying fractions, multiply the numerators together and the denominators together.

Example:

An engineer needs 1/3 of a 1/2 liter bottle of sealant:

So, 1/6 liter of sealant is needed.

22. Division of Fractions

To divide fractions, multiply the first fraction by the reciprocal of the second fraction.

Example:

If an engineer has 3/4 liter of fluid and each application uses 1/2 liter:

This means 1 1/2 applications are possible with the fluid available.

23. Addition of Mixed Numbers

To add mixed numbers, add the whole numbers separately from the fractions. Convert any improper fractions if needed.

Example:

If a maintenance log shows 2 1/4 hours spent on one task and 1 3/4 hours on another:

Add the whole numbers:
Add the fractions:

So, the total time spent is:

24. Subtraction of Mixed Numbers

When subtracting mixed numbers, subtract the whole numbers and fractions separately. Borrow from the whole number if needed.

Example:

If a task was estimated to take 3 1/2 hours but only took 2 3/4 hours:

Subtract the whole numbers:
Subtract the fractions:

Convert 1/2 to 2/4:

So, the actual time difference is 1 – 1/4 = 3/4 hour less than estimated.

25. The Decimal Number System

The decimal system is a base-10 numbering system, using digits from 0 to 9. Each position in a number represents a power of 10, essential in aviation for accurate measurements, calculations, and data logging.

Example:

The value 345.67 represents $3 \times 100 + 4 \times 10 + 5 \times 1 + 6 \times 0.1 + 7 \times 0.01$ , or 345.67.

26. Origin and Definition

Decimals originate from the base-10 system, commonly used for calculations requiring precision. In aviation, decimal accuracy is crucial for fuel measurements, tolerances, and part dimensions.

Example:

Fuel may be measured as 5.75 liters to ensure precise tracking, where .75 means three-quarters of a liter.

27. Addition of Decimal Numbers

Align decimal points before adding. In aviation, decimal addition is useful for totaling maintenance hours or fuel quantities.

Example:

If two maintenance tasks take 2.35 and 1.45 hours, the total time is:

28. Subtraction of Decimal Numbers

Align decimal points before subtracting. Decimal subtraction is used in aviation to calculate differences, such as fuel remaining after usage.

Example:

If an aircraft starts with 8.50 liters and uses 3.25 liters:

29. Multiplication of Decimal Numbers

Multiply as whole numbers, then place the decimal in the result. Decimal multiplication helps calculate quantities like lubricant needed for multiple parts.

Example:

If each part requires 0.75 liters and there are 4 parts:

30. Division of Decimal Numbers

To divide decimals, shift the decimal in the divisor to make it a whole number, then divide. Decimal division helps allocate resources like fuel or oil in aviation maintenance.

Example:

If 6.3 liters of oil is divided among 3 engines:

31. Rounding Off Decimal Numbers

Rounding is reducing a decimal to a simpler value, often based on significant figures or practical limits in aviation.

Example:

A fuel measurement of 7.256 liters can be rounded to 7.26 liters for practicality.

32. Converting Decimal Numbers to Fractions

Convert a decimal to a fraction by identifying place value and simplifying. This conversion is useful when documentation requires fractional measurements.

Example:

The decimal 0.75 converts to 3/4 by recognizing it as 75/100 and simplifying.

33. Decimal Equivalent Chart

A decimal equivalent chart lists common decimal and fraction pairs, providing quick reference during measurements and conversions in aviation maintenance.

Example:

0.5 = 1/2, 0.25 = 1/4, 0.75 = 3/4—these help quickly convert between decimals and fractions when reading gauges or tools.

34. Ratio

A ratio is a comparison between two quantities, showing how much of one there is relative to the other. Ratios are widely used in aviation to compare fuel to distance, parts specifications, or weight distributions.

Example:

A fuel ratio of 3:1 means that for every 3 parts of fuel in Tank A, there is 1 part in Tank B.

35. Aviation Applications of Ratio

In aviation, ratios help maintain safe operational limits, such as fuel-to-weight ratios, mixture ratios, and tire pressure ratios. Properly managing these ratios ensures optimal performance and safety.

Example:

Compression ratio on a reciprocating engine is the ratio of the volume of a cylinder with the piston at the bottom of its stroke to the volume of the cylinder with the piston at the top of its stroke. For example, a typical compression ratio might be 10:1 (or 10 to 1).

Aspect ratio is the ratio of the length (or span) of an airfoil to its width (or chord). A typical aspect ratio for a commercial airliner might be 7:1 (or 7 to 1).

Air-fuel ratio is the ratio of the weight of the air to the weight of fuel in the mixture being fed into the cylinders of a reciprocating engine. For example, a typical air-fuel ratio might be 14.3:1 (or 14.3 to 1).

Gear ratio is the number of teeth each gear represents when two gears are used in an aircraft component. In the figure, the pinion gear has 8 teeth and a spur gear has 28 teeth. The gear ratio is 8:28. Using 7 as the LCD, 8:28 becomes 2:7.

36. Proportion

A proportion is an equation stating that two ratios are equal. In aviation, proportions are used to scale measurements, allocate resources, or maintain balance across multiple components.

Example:

If an aircraft uses 10 gallons of fuel per 200 miles, this can be written as a proportion to calculate the fuel required for 500 miles:

37. Solving Proportions

To solve proportions, cross-multiply and divide to find the unknown value. This method is useful in aviation to determine fuel, speed, or weight limits.

Example:

Using the proportion above, we find the fuel needed for 500 miles:

So, 25 gallons of fuel are required for 500 miles.

38. Average Value

The average, or mean, is calculated by summing a set of values and dividing by the number of values. In aviation maintenance, averages help in estimating component wear rates, maintenance times, or fuel consumption.

Example:

An engineer records fuel consumption for five flights: 50, 60, 55, 65, and 70 liters. The average fuel consumption is:

So, the average fuel consumption per flight is 60 liters.

39. Percentage

A percentage represents a part per hundred and is used in aviation for performance metrics, efficiency ratings, and inspection criteria.

Example:

If an aircraft’s battery life has decreased to 80% of its original capacity, this means the battery can operate at 80% of its original efficiency.

40. Expressing a Decimal Number as a Percentage

To convert a decimal to a percentage, multiply by 100 and add a percent symbol (%). This conversion is helpful when representing efficiency, completion rates, or performance metrics in aviation.

Example:

A gauge shows 0.75 of its maximum pressure. Converting 0.75 to a percentage:

So, the pressure is at 75% of its maximum.

41. Expressing a Percentage as a Decimal Number

To convert a percentage to a decimal, divide by 100. This conversion simplifies calculations in aviation where percentages need to be used in formulas or equations.

Example:

An engineer finds that 20% of a fuel tank’s capacity is reserved for safety. Converting 20% to a decimal:

42. Expressing a Fraction as a Percentage

To convert a fraction to a percentage, divide the numerator by the denominator and multiply by 100.

Example:

A maintenance log shows 3/4 of tasks are complete. Converting 3/4 to a percentage:

This means 75% of tasks are completed.

43. Finding a Percentage of a Given Number

To find a percentage of a number, multiply the number by the percentage (expressed as a decimal). This calculation is common in aviation when determining fuel reserves, efficiency, or load limits.

Example:

If an aircraft’s fuel tank holds 200 liters and 25% is reserved for emergencies:

So, 50 liters of fuel are reserved for emergencies.

44. Finding What Percentage One Number is of Another

To find what percentage one number is of another, divide the part by the whole and multiply by 100. This is useful in aviation to measure efficiency, usage rates, or adherence to limits.

Example:

If 15 hours of maintenance were logged out of 60 hours available:

This means 25% of the available maintenance time was used.

45. Finding a Number When a Percentage of it is Known

To find a number when you know a percentage of it, divide the known percentage by its decimal equivalent. This is useful for calculating total quantities, capacities, or limits based on partial information.

Example:

If 30 liters is 40% of the maximum hydraulic fluid capacity, find the total capacity:

So, the maximum capacity is 75 liters.

46. Powers and Indices

A power or index shows how many times a number is multiplied by itself. Powers and indices simplify large calculations, which are frequent in aviation for measurements, specifications, and force calculations.

Example:

If a force is doubled three times, we express this as

47. Squares and Cubes

The square of a number (raised to the power of 2) represents an area, while the cube (raised to the power of 3) represents a volume. In aviation, squares are used in area calculations for wing surfaces, and cubes in volume measurements for fuel or cargo.

Example:

The square of 5 (5²) is , while the cube of 3 (3³) is

48. Negative Powers

A negative power represents the reciprocal of a positive power. Negative powers simplify handling very small quantities, often seen in aviation for precision measurements and tolerances.

Example:

49. Law of Exponents

The law of exponents helps simplify calculations by applying rules like multiplying powers, dividing powers, and raising a power to another power. In aviation, these rules streamline calculations in formulas involving force, pressure, and energy.

Example:

Using the product rule,

50. Powers of Ten

Powers of ten are used to express large or small quantities in a manageable form, essential in aviation for measurements such as engine thrust (thousands of Newtons) or pressure tolerances (micronewtons).

Example:

51. Roots

The root of a number is a value that, when multiplied by itself a certain number of times, gives the original number. Roots are essential in aviation for calculations involving forces, pressures, and engineering tolerances.

Example:

The square root of 16 is 4 because Similarly, the cube root of 27 is 3 because .

52. Square Roots

The square root of a number is a value that, when squared, returns the original number. Square roots are used in aviation when calculating load distributions, structural stress, and pressure.

Example:

If an engineer needs to determine the stress load on a square-shaped area of material, they may need to find the square root of the area. For an area of 36 square meters, the side length is:

53. Cube Roots

The cube root of a number is a value that, when cubed, equals the original number. Cube roots are applied in aviation for volume calculations, especially when scaling fuel, cargo spaces, or engine compartments.

Example:

If a fuel tank has a volume of 125 cubic meters, the side length (if it were a perfect cube) is:

54. Fractional Indices

Fractional indices represent roots in exponential form, simplifying complex root calculations. In aviation, fractional indices are used in formulas where roots of specific powers are needed for precise measurements or design specifications.

Example:

55. Scientific and Engineering Notation

Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10, simplifying very large or small values. Engineering notation is similar but uses powers of 10 in multiples of three, aligning with metric prefixes, making it practical in aviation for measurements like distances, weights, and forces.

Example:

56. Converting Numbers from Standard Notation to Scientific or Engineering Notation

To convert a standard number, move the decimal point until one digit remains on the left (for scientific) or to the nearest thousand, million, etc. (for engineering), adjusting the power of 10 accordingly. This is common in aviation for recording specifications like thrust or fuel capacity.

Example:

57. Converting Numbers from Scientific or Engineering Notation to Standard Notation

To convert back, move the decimal point based on the exponent’s value in scientific notation or the metric prefix in engineering notation.

Example:

58. Addition, Subtraction, Multiplication, Division of Scientific and Engineering Numbers

Addition/Subtraction: Ensure numbers have the same exponent before adding or subtracting the significant figures.
Multiplication: Multiply the significant figures and add the exponents.
Division: Divide the significant figures and subtract the exponents.

Example:

59. Denominated Numbers

Denominated numbers are numbers with specific units attached, such as hours, kilograms, or liters. In aviation, denominated numbers are essential for measurements of fuel, weight, distance, and time, allowing for precise calculations and tracking.

Example:

A fuel load of 200 liters is a denominated number, as it specifies both the quantity and the unit of measure.

60. Addition of Denominated Numbers

When adding denominated numbers, the units must match. Only quantities with the same unit can be combined, which is crucial in aviation to avoid errors in fuel calculations, weights, or time logs.

Example:

If an aircraft is loaded with 100 liters of fuel and then receives 150 liters more:

61. Subtraction of Denominated Numbers

Subtraction with denominated numbers also requires matching units. This is common when calculating remaining fuel, weight reduction, or time left on a flight.

Example:

If an aircraft starts with 500 liters of fuel and uses 300 liters during a flight:

So, 200 liters of fuel remain.

62. Multiplication of Denominated Numbers

Multiplying denominated numbers is used to calculate area, volume, or total quantities based on a rate. The resulting unit depends on the units involved in the multiplication.

Example:

If an aircraft requires 3 liters of hydraulic fluid per system and has 4 systems:

63. Division of Denominated Numbers

Division with denominated numbers is used to calculate rates or unit conversions. This is crucial for calculating consumption rates or distributing quantities across units in aviation.

Example:

If an aircraft has 600 kg of cargo divided across 3 compartments:

64. Area and Volume

Area measures the space within a shape’s boundaries, often applied to surfaces like wings, while volume measures the space a 3D object occupies, used for tanks and cargo compartments.

Example:

Calculating the wing area of an aircraft helps determine lift capacity, while the fuel tank volume indicates fuel-carrying capacity.

65. Rectangle

The area of a rectangle is calculated by multiplying its length by its width. This applies to surfaces like aircraft panels or cargo floors.

Example:

If an aircraft panel measures 5 meters by 2 meters:

66. Square

The area of a square, with equal-length sides, is the side length squared. This is useful for square-shaped instrument panels or structural sections.

Example:

If each side of a square panel is 3 meters:

67. Triangle

A triangle is a three-sided figure. The sum of the three angles in a triangle is always equal to 180°. Triangles are often classified by their sides (shown in figure below).

The area of a triangle is half the base times the height. Triangular shapes are found in stabilizers and wing supports.

Example:

If a support in a stabilizer is 4 meters at the base and 3 meters high:

68. Parallelogram

The area of a parallelogram is base times height. In aviation, this shape might represent angled panels.

Example:

For a parallelogram panel with a base (length) of 6 meters and height of 2 meters:

69. Trapezoid

The area of a trapezoid is the average of the two parallel sides (bases) times the height.

Example:

A panel has bases of 5 meters and 3 meters and a height of 2 meters:

70. Circle

The area of a circle is calculated using π (pi) times the radius squared, essential for calculating wheel or hatch areas.

Example:

If an aircraft hatch has a radius of 1 meter:

71. Ellipse

The area of an ellipse is π times the product of the semi-major and semi-minor axes.

Example:

For an elliptical window with a semi-major axis of 2 meters and semi-minor axis of 1 meter:

72. Wing Area

To describe a wing’s shape and calculate its area, two main terms are needed: “span” and “chord.” The span (S) is the distance from one wingtip to the other, while the chord is the wing’s average width from the front (leading edge) to the back (trailing edge). For tapered wings, the mean chord (C) represents this average width. The wing area can then be calculated using:

Example:

Suppose an aircraft wing has:

Span (S) = 30 meters
Mean Chord (C) = 5 meters

Wing Area = 30 meters × 5 meters = 150 square meters

73. Volume

Volume is the amount of space an object occupies, measured in cubic units. In aviation, volume calculations are essential for determining fuel tank capacity, cargo space, and the size of hydraulic fluid reservoirs.

74. Rectangular Solids

The volume of a rectangular solid is found by multiplying its length, width, and height. This applies to box-shaped components like cargo compartments and storage containers.

Example:

If a cargo compartment has dimensions 4 meters (length), 3 meters (width), and 2 meters (height):

The compartment has a volume of 24 cubic meters.

75. Cube

A cube is a rectangular solid with equal-length sides. To find the volume, cube (raise to the third power) one of the sides.

Example:

If a container measures 3 meters on each side:

The volume of the container is 27 cubic meters.

76. Cylinder

The volume of a cylinder is found by multiplying the area of the base (a circle) by the cylinder’s height. This is commonly used for fuel tanks and hydraulic reservoirs.