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3.4.1: What Are Percentages?


Lesson

Let's learn about percentages.

Exercise (PageIndex{1}): Dollars and Cents

Find each answer mentally.

  1. A sticker costs 25 cents. How many dollars is that?
  2. A pen costs 1.50 dollars. How many cents is that?
  3. How many cents are in one dollar?
  4. How many dollars are in one cent?

Exercise (PageIndex{2}): Coins

  1. Complete the table to show the values of these U.S. coins.
coinpennynickeldimequarterhalf dollardollar
value (cents)
Table (PageIndex{1})

The value of a quarter is 25% of the value of a dollar because there are 25 cents for every 100 cents.

  1. Write the name of the coin that matches each expression.
    • 25% of a dollar
    • 5% of a dollar
    • 1% of a dollar
    • 100% of a dollar
    • 10% of a dollar
    • 50% of a dollar
  2. The value of 6 dimes is what percent of the value of a dollar?
  3. The value of 6 quarters is what percent of the value of a dollar?

Are you ready for more?

Find two different sets of coins that each make 120% of a dollar, where no type of coin is in both sets.

Exercise (PageIndex{3}): Coins on a Number Line

A $1 coin is worth 100% of the value of a dollar. Here is a double number line that shows this.

  1. The coins in Jada’s pocket are worth 75% of a dollar. How much are they worth (in dollars)?
  2. The coins in Diego’s pocket are worth 150% of a dollar. How much are they worth (in dollars)?
  3. Elena has 3 quarters and 5 dimes. What percentage of a dollar does she have?

Summary

A percentage is a rate per 100.

We can find percentages of $10 using a double number line where 10 and 100% are aligned, as shown here:

Looking at the double number line, we can see that $5.00 is 50% of $10.00 and that $12.50 is 125% of $10.00.

Glossary Entries

Definition: Percent

The word percent means “for each 100.” The symbol for percent is %.

For example, a quarter is worth 25 cents, and a dollar is worth 100 cents. We can say that a quarter is worth 25% of a dollar.

Definition: Percentage

A percentage is a rate per 100.

For example, a fish tank can hold 36 liters. Right now there is 27 liters of water in the tank. The percentage of the tank that is full is 75%.

Practice

Exercise (PageIndex{4})

What percentage of a dollar is the value of each coin combination?

  1. (4) dimes
  2. (1) nickel and (3) pennies
  3. (5) quarters and (1) dime

Exercise (PageIndex{5})

  1. List three different combinations of coins, each with a value of 30% of a dollar.
  2. List two different combinations of coins, each with a value of 140% of a dollar.

Exercise (PageIndex{6})

The United States government used to make coins of many different values. For each coin, state its worth as a percentage of $1.

(frac{1}{2} ext{ cent }qquad 3 ext{ cents }qquad 20 ext{ cents }qquad $2frac{1}{2}qquad $5)

Exercise (PageIndex{7})

Complete the double number to line show percentages of $50.

Exercise (PageIndex{8})

Elena bought 8 tokens for $4.40. At this rate:

  1. How many tokens could she buy with $6.05?
  2. How much do 19 tokens cost?

(From Unit 3.3.5)

Exercise (PageIndex{9})

A snail travels 10 cm in 4 minutes. At this rate:

  1. How long will it take the snail to travel 24 cm?
  2. How far does the snail travel in 6 minutes?

(From Unit 3.3.4)

Exercise (PageIndex{10})

  1. 3 tacos cost $18. Complete the table to show the cost of 4, 5, and 6 tacos at the same rate.
    number of tacoscost in dollarsrate in dollars per taco
    (3)(18)
    (4)
    (5)
    (6)
    Table (PageIndex{2})
  2. If you buy (t) tacos for (c) dollars, what is the unit rate?

(From Unit 3.3.3)


3.4.1: What Are Percentages?

Mixing Ratio Reference Material Overview
Return

Mixing ratios can be stated either in parts or percentages, parts (normally by volume) being the most commonly used and easiest to understand.

When a mixing ratio is given in parts, the measurement chosen as one part can vary greatly. For example, One teaspoon could be used to measure one part, or one 55 gallon steel drum could be used to measure one part. Regardless of size, the chosen measurement must remain the same throughout the mixing process. A mixing ratio given as 4:2:1 normally means 4 parts of base product, 2 parts thinner/reducer, and 1 part hardener. However, some paint manufactures add hardener second, and thinner/reducer last.

When a mixing ratio is given as a percentage, convert the percentage to a fraction, then think of the fraction as parts solvent/parts paint. Examples of this type of mixing ratio is best illustrated below on the . For instance, the 33% listed in the chart would be 1/3 or one part solvent (thinner/reducer) to three parts paint.

Reduction Recommendations Chart
Return

PERCENTAGE FRACTION AMOUNT OF THINNER / REDUCER AMOUNT OF PAINT
25% 1/4 1 part 4 part
33% 1/3 1 part 3 part
50% 1/2 1 part 2 part
75% 3/4 3 part 4 part
100% 1/1 1 part 1 part
125% 5/4 5 part 4 part
150% 3/2 3 part 2 part
200% 2/1 2 part 1 part

Reduction percentages can become confusing. Areas where problems may arise include the following:

    50-50 may mean 50% to the painter, but in reality is 1 to 1 or 100% reduction.


Paint Reduction Equivalent Chart
Return

Percent Reducer or Thinner
Percent Paint Reducer or Thinner Paint Reducer or Thinner
12 - 1/2% 8 1 8 oz. 1 oz.
25% 4 1 8 oz. 2 oz.
33 - 1/3% 3 1 8 oz. 2 - 3/5 oz.
40% 5 2 8 oz. 3 - 1/5 oz.
50% 2 1 8 oz. 4 oz.
75% 4 3 8 oz. 6 oz.
100% 1 1 8 oz. 8 oz.
125% 4 5 8 oz. 10 oz.
150% 2 3 8 oz. 12 oz.
200% 1 2 8 oz. 16 oz.
12 - 1/2% 8 1 16 oz. 2 oz.
25% 4 1 16 oz. 4 oz.
33 - 1/3% 3 1 16 oz. 5 - 1/3 oz.
40% 5 2 16 oz. 6 - 2/5 oz.
50% 2 1 16 oz. 8 oz.
75% 4 3 16 oz. 12 oz.
100% 1 1 16 oz. 16 oz.
125% 4 5 16 oz. 20 oz.
150% 2 3 16 oz. 24 oz.
200% 1 2 16 oz. 32 oz.
12 - 1/2% 8 1 32 oz. 4 oz.
25% 4 1 32 oz. 8 oz.
33 - 1/3% 3 1 32 oz. 10 - 3/5 oz.
40% 5 2 32 oz. 12 - 4/5 oz.
50% 2 1 32 oz. 16 oz.
75% 4 3 32 oz. 24 oz.
100% 1 1 32 oz. 32 oz.
125% 4 5 32 oz. 40 oz.
150% 2 3 32 oz. 48 oz.
200% 1 2 32 oz. 64 oz.
12 - 1/2% 8 1 1 gallon 1 pint
25% 4 1 1 gallon 1 quart
33 - 1/3% 3 1 1 gallon 1 - 1/3 quarts
40% 5 2 1 gallon 3 pints
50% 2 1 1 gallon 2 quarts
75% 4 3 1 gallon 3 quarts
100% 1 1 1 gallon 4 quarts
125% 4 5 1 gallon 5 quarts
150% 2 3 1 gallon 6 quarts
200% 1 2 1 gallon 8 quarts


Measuring and Mixing utilizing a Mixing Stick (Volume)
Return

    Insert mixing stick into the container
    and hold vertically.


Measuring and Mixing utilizing a Graduated Container (Volume)
Return


Abstract

Purpose

Gleason grading is an important predictor of prostate cancer (PCa) outcomes. Studies using surrogate PCa end points suggest outcomes for Gleason score (GS) 7 cancers vary according to the predominance of pattern 4. These studies have influenced clinical practice, but it is unclear if rates of PCa mortality differ for 3 + 4 and 4 + 3 tumors. Using PCa mortality as the primary end point, we compared outcomes in Gleason 3 + 4 and 4 + 3 cancers, and the predictive ability of GS from a standardized review versus original scoring.

Patients and Methods

Three study pathologists conducted a blinded standardized review of 693 prostatectomy and 119 biopsy specimens to assign primary and secondary Gleason patterns. Tumor specimens were from PCa patients diagnosed between 1984 and 2004 from the Physicians' Health Study and Health Professionals Follow-Up Study. Lethal PCa (n = 53) was defined as development of bony metastases or PCa death. Hazard ratios (HR) were estimated according to original GS and standardized GS. We compared the discrimination of standardized and original grading with C-statistics from models of 10-year survival.

Results

For prostatectomy specimens, 4 + 3 cancers were associated with a three-fold increase in lethal PCa compared with 3 + 4 cancers (95% CI, 1.1 to 8.6). The discrimination of models of standardized scores from prostatectomy (C-statistic, 0.86) and biopsy (C-statistic, 0.85) were improved compared to models of original scores (prostatectomy C-statistic, 0.82 biopsy C-statistic, 0.72).

Conclusion

Ignoring the predominance of Gleason pattern 4 in GS 7 cancers may conceal important prognostic information. A standardized review of GS can improve prediction of PCa survival.


ACI 318-08 Seismic Provisions

ACI 318-08 Appendix D seismic design consists of three options defined by the provisions given in Part D.3.3.4, Part D.3.3.5 and Part D.3.3.6. The provisions in the option selected must be satisfied for both tension and shear load conditions.

ACI 318-08 Appendix D seismic design criteria can be summarized as follows:

  • calculate nominal strengths corresponding to possible anchor failure modes per Part D.4.1.
  • apply a strength reduction factor (-factor) to each nominal strength per Part D.4.1.2.
  • apply a seismic reduction factor of 0.75 to non-steel design strengths per Part D.3.3.3.

The commentary RD.3.3.3 notes that the 0.75 factor is applied “to account for increased damage states in the concrete resulting from seismic actions.” When the design is controlled by non-ductile anchor strengths, an additional reduction factor must be applied to the calculated anchor design strengths corresponding to brittle failure modes. This criterion will be covered when discussing Part D.3.3.6.

Part D.3.3.4 can be used if the anchorage design is governed by the steel strength of a ductile steel element. The design steel strength in tension, defined by the parameter Nsa, must be the controlling tension design strength compared to the non-steel tension design strengths defined by the parameter (0.75)(NN). Likewise, the design steel strength in shear, defined by the parameter Vsa, must be the controlling shear design strength compared to the non-steel shear design strengths defined by the parameter (0.75)(VN). Part D.1 – Definitions defines a ductile steel element as having a tensile test elongation of at least fourteen percent measured over a specified gauge length, and a reduction in cross-sectional area of at least thirty percent. Anchor elements that do not satisfy these criteria, or for which these criteria are not determined, are assumed to be brittle steel elements, which precludes them from design with the provisions of D.3.3.4.

Part D.3.3.5 can be used if the anchorage design is controlled by ductile yielding of the attachment. The force calculated to yield the attachment must be less than or equal to the calculated anchor design strengths. Tension anchor design strengths are defined as Nsa for steel failure and (0.75)(NN) for non-steel failure. Shear anchor design strengths are defined by Vsa for steel failure and (0.75)(VN) for non-steel failure.

Part D.3.3.4 and Part D.3.3.5 are both predicated on a ductile failure mode controlling the anchorage design. The ACI 318 code recognizes, however, that an anchorage design controlled by a ductile failure mode may not be possible. For example, anchor spacing and edge distance, concrete member thickness, or base plate properties may preclude an anchorage design controlled by a ductile failure mode. Therefore, Part D.3.3.6 provides another option that waives any ductility requirement and permits the anchorage design to be controlled by a brittle failure mode. The provisions of Part D.3.3.6 include an additional reduction factor which must be applied to anchor design strengths corresponding to brittle failure modes. For simplicity, this factor will be referred to in this article as nonductile.

nonductile is applied to non-steel anchor design strengths (nonductile 0.75 NN and nonductile 0.75 VN) as well as to steel design strengths for anchor elements considered to be brittle (nonductile steel Nsa and nonductile steel Vsa). The default value for nonductile is 0.4 however, it can vary depending on the design conditions being considered. Part D.3.3.6 notes that a nonductile value of 0.5 can be used for “anchors of stud bearing walls” because this application typically consists of multiple anchors capable of load redistribution. The 2009 IBC Section 1908.1.9 waives the use of nonductile for anchorage of nonstructural components and anchors designed to resist wall out-of-plane forces. Figure 1 summarizes ACI 318-08 Appendix D seismic calculations.


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Total body surface burned

The burn percentage is estimated according to the Wallace rule of nines on body surface area .

This is a calculation adapted for both adults and children and adds percentages according to the body parts burned in order to deliver a final percentage that is then used in estimating the severity of the burns.

The burn percentage is then employed in the Parkland formula to determine the fluid requirement to be replaced in the first 24h.

The following table defines the Wallace rule of nines:

Region Adults Children
head 9% 18%
front torso 18% 18%
back torso 18% 18%
arms 9% for each 9% for each
legs 9% for each 14% for each


5 Transposon-induced Mutations and Inserts

Three types of genetic inserts are involved in creating transposon-induced mutations. Two lines, one carrying the transposable-element as a concatamer and the other carrying the transposase are mated. This causes the transposable-element to come in contact with the transposase and to be mobilized from its original site, and, when reintegrated into the genome, can cause a heritable phenotypic mutation. (c.f., Ding, et al.,2005 Bestor, 2005 Dupuy, et al., 2005). Accepted nomenclature for the transposable-element inserts, transposase transgenes, and resulting transposed insertion alleles are given below.

5.1 Transgenic Transposable Element (TE) Concatamers

The transgenic transposable element concatamers are identified with a standard prefix Tg (for transgenic) and Tn (for transposable element). The class of transposable element may be included in parentheses. The general format of the symbol is:

  • Tg denoting transgenic
  • Tn denoting transposon
  • In parentheses, a lowercase abbreviation of the transposon class (in this case sb for Sleeping Beauty), followed by a hyphen and the vector designation
  • The laboratory's line or founder designation or a serial number
  • The Laboratory Code of the originating lab

5.2 Transposase Inserts

Transposases can be engineered into the genome via transgenesis or specific gene targeting. In these cases the relevant nomenclature for transgenes or targeted mutations is used.

For a transgene, use the standard prefix Tg (for transgene). The contents of the parentheses will usually be the promoter and the symbol for the transposase with which it is associated, separated by a hyphen. The general format of the symbol is:

  • Tg denoting transgene
  • In parentheses, the official gene symbol for the promoter, using the nomenclature of the species of origin, followed by a hyphen and a lowercase transposase symbol, in this case sb10 for the Sleeping Beauty 10 transposase
  • The laboratory's line or founder designation or a serial number
  • The Laboratory code of the originating lab

For a targeted knock-in of the transposase, use the standard format for a targeted mutation, i.e., the symbol of the targeted gene with a superscripted allele symbol beginning with the prefix tm. The contents of the parentheses will usually be the symbol for the transposase with which it is associated. The general format of the symbol is:

  • The gene into which the transposase was integrated, in this case Gt(ROSA)26Sor
  • In the superscript:
    • tm denoting targeted mutation
    • A serial number of the targeted mutation
    • In parentheses, a lowercase transposase symbol, in this case sb11 for the Sleeping Beauty 11 transposase
    • The Laboratory Code of the originating lab

    5.3 Transposed Insertion Alleles

    These alleles follow the rules for naming all other alleles. In general a transposable element concatamer marker will already be established, as above. The new allele, then, will be a superscripted form of the concatamer symbol. Note that all such alleles that are "derived from" a transposable element concatamer carry the original number with a decimal point and serial number identifying the specific allele. The general format is:

    • The gene into which the transposable element was integrated (transposed)
    • In the superscript:
      • Tn denoting transposon
      • In parentheses, a lowercase abbreviation of the transposon class (in this case sb for Sleeping Beauty), followed by a hyphen and the vector designation
      • A serial number, in which the primary number corresponds to that given to the transposable element concatamer from which it arose, followed by a decimal point and a serial number designating its number within the series of derivative insertion alleles.
      • The Laboratory Code of the lab originating the transposable element line

      If a newly transposed insertion occurs in an unknown site or intergenic region, the form:

      is used to symbolize the "genomic mutation" without being superscripted to a gene symbol, similar to the way a random transgene inserted into a non-gene site is designated.


      Flow Formulas & Cv Factors

      Valves with a "Full Port" have an internal seat diameter that is the same as the nominal pipe size, i.e. a 1 inch pipe size valve with a full port has a 1 inch diameter seat. Valves with a "Reduced Port" have an internal seat diameter that is smaller than the nominal pipe size.

      The valve's flow coefficient, Cv, is a value that is determined by flow testing for each valve size. Full port valves will have a higher Cv than reduced port valves. The Cv rating for each valve is listed in the tables found in the valve catalog.

      The definition of Cv is the # of gallons of water that will flow through the valve with a 1 PSI pressure differential when the valve is open.

      The equations below can be used to determine:

      • Flow Rate, given the Cv and &DeltaP
      • Cv, given the Flow Rate and &DeltaP
      • &DeltaP, given the Flow Rate and Cv

      Cv = Valve's flow coefficient (dimensionless value)
      S = Specific Gravity (1.0 for air or water)
      T = Absolute Temperature in °R (°R = °F + 460)
      P 1 = Inlet Pressure in PSIG
      &DeltaP = Pressure Differential in PSI across valve in the open position
      V = Specific Volume in Cubic Feet per Pound


      3.3.1. The scoring parameter: defining model evaluation rules¶

      Model selection and evaluation using tools, such as model_selection.GridSearchCV and model_selection.cross_val_score , take a scoring parameter that controls what metric they apply to the estimators evaluated.

      3.3.1.1. Common cases: predefined values¶

      For the most common use cases, you can designate a scorer object with the scoring parameter the table below shows all possible values. All scorer objects follow the convention that higher return values are better than lower return values. Thus metrics which measure the distance between the model and the data, like metrics.mean_squared_error , are available as neg_mean_squared_error which return the negated value of the metric.

      Classification

      requires predict_proba support

      suffixes apply as with ‘f1’

      suffixes apply as with ‘f1’

      suffixes apply as with ‘f1’

      The values listed by the ValueError exception correspond to the functions measuring prediction accuracy described in the following sections. The scorer objects for those functions are stored in the dictionary sklearn.metrics.SCORERS .

      3.3.1.2. Defining your scoring strategy from metric functions¶

      The module sklearn.metrics also exposes a set of simple functions measuring a prediction error given ground truth and prediction:

      functions ending with _score return a value to maximize, the higher the better.

      functions ending with _error or _loss return a value to minimize, the lower the better. When converting into a scorer object using make_scorer , set the greater_is_better parameter to False ( True by default see the parameter description below).

      Metrics available for various machine learning tasks are detailed in sections below.

      Many metrics are not given names to be used as scoring values, sometimes because they require additional parameters, such as fbeta_score . In such cases, you need to generate an appropriate scoring object. The simplest way to generate a callable object for scoring is by using make_scorer . That function converts metrics into callables that can be used for model evaluation.

      One typical use case is to wrap an existing metric function from the library with non-default values for its parameters, such as the beta parameter for the fbeta_score function:

      The second use case is to build a completely custom scorer object from a simple python function using make_scorer , which can take several parameters:

      the python function you want to use ( my_custom_loss_func in the example below)

      whether the python function returns a score ( greater_is_better=True , the default) or a loss ( greater_is_better=False ). If a loss, the output of the python function is negated by the scorer object, conforming to the cross validation convention that scorers return higher values for better models.

      for classification metrics only: whether the python function you provided requires continuous decision certainties ( needs_threshold=True ). The default value is False.

      any additional parameters, such as beta or labels in f1_score .

      Here is an example of building custom scorers, and of using the greater_is_better parameter:

      3.3.1.3. Implementing your own scoring object¶

      You can generate even more flexible model scorers by constructing your own scoring object from scratch, without using the make_scorer factory. For a callable to be a scorer, it needs to meet the protocol specified by the following two rules:

      It can be called with parameters (estimator, X, y) , where estimator is the model that should be evaluated, X is validation data, and y is the ground truth target for X (in the supervised case) or None (in the unsupervised case).

      It returns a floating point number that quantifies the estimator prediction quality on X , with reference to y . Again, by convention higher numbers are better, so if your scorer returns loss, that value should be negated.

      Using custom scorers in functions where n_jobs > 1

      While defining the custom scoring function alongside the calling function should work out of the box with the default joblib backend (loky), importing it from another module will be a more robust approach and work independently of the joblib backend.

      For example, to use n_jobs greater than 1 in the example below, custom_scoring_function function is saved in a user-created module ( custom_scorer_module.py ) and imported:

      3.3.1.4. Using multiple metric evaluation¶

      Scikit-learn also permits evaluation of multiple metrics in GridSearchCV , RandomizedSearchCV and cross_validate .

      There are three ways to specify multiple scoring metrics for the scoring parameter:

        As an iterable of string metrics::

      Note that the dict values can either be scorer functions or one of the predefined metric strings.

      As a callable that returns a dictionary of scores:


      3.4.1: What Are Percentages?

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