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Choose the correct base, combine rates and distinguish totals from parts.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
Translate a ratio, rate or percent situation into a unit-consistent equation and compute the exact requested quantity.
Review fraction multiplication, division and conversion to decimals. Three-fifths equals 0.6 and sixty percent; these are representations of the same number. Dividing by a fraction measures how many portions of that size fit into a quantity. If a worker completes one quarter of a job per hour, the time for one job is one divided by one quarter, or four hours. Keep the fraction until a decimal is useful, especially when rounding could accumulate.
| Term | What it means |
|---|---|
| Base | The reference quantity used as the denominator of a percent or ratio. |
| Rate | An amount per unit of another quantity, such as kilometers per hour. |
| Weighted mean | A mean that accounts for how much each component contributes. |
| Multiplier | A factor converting an old value directly into a new value. |
A ratio compares quantities, a rate compares quantities with different units, and a percent expresses a ratio per hundred. In all three, identifying the denominator is more important than performing the final arithmetic. Thirty percent of an original price and thirty percent of a discounted price are generally different amounts. A red-to-blue ratio is not the fraction of all tiles that are red. Average speed compares total distance with total elapsed time, not one speed with another.
Begin by naming the quantities and their units. Then write the relationship before entering numbers into a calculator. A short equation exposes a mistaken base more clearly than a long chain of unexplained arithmetic. After calculating, compare the result with simple bounds: a discounted price should be smaller, a mixture concentration should lie between the component concentrations, and a combined positive work rate should exceed either individual rate.
Another way: steps
Name the quantities and units; identify the base; write the model; calculate; check the requested interpretation.
An increase of p percent changes a value x to x(1 + p/100). A decrease uses x(1 - p/100). Apply successive changes by multiplying their factors in order. Increasing by twenty percent and then decreasing by twenty percent multiplies the original by 1.2 times 0.8 = 0.96, a net decrease of four percent. The changes do not cancel because the second percentage uses the enlarged value. Distinguish a percent change from a percentage-point change: a rate rising from ten percent to twelve percent rises two percentage points but twenty percent relative to its original value.
If an item costs ninety dollars after a ten percent discount, ninety is ninety percent of the original. Write 0.9x = 90, giving x = 100. Adding ten percent of ninety gives ninety-nine and fails to reverse the discount because it uses the wrong base. The same principle applies to tax-inclusive prices, remaining populations and losses. Draw an arrow from original to final labeled with the multiplier; reverse the arrow by dividing by that same multiplier. Verify the original by applying the stated change once more.
If boys and girls are in the ratio two to three, represent their counts by 2k and 3k. The total is 5k, so boys form two-fifths of the group. Treating two-thirds as the fraction of all students confuses a part-to-part comparison with a part-to-whole fraction. If the difference in counts is twelve, then k = 12 because 3k - 2k = k. If the total is sixty, k = 12 for a different reason. Translate the given total, difference or individual count explicitly instead of mechanically adding or subtracting the ratio entries.
Two pumps filling the same tank have rates measured in tanks per hour, so those rates can be added. Their completion times cannot be added to obtain a joint completion time. A drain has a negative contribution if filling is defined as positive. A pump filling one sixth of a tank per hour and a drain removing one tenth per hour produce a net rate of one fifteenth of a tank per hour. Starting from empty, the tank fills in fifteen hours if the rates remain constant. State that assumption; real systems may have changing rates, but these mathematical models specify or imply constant rates.
For a trip, add distances and add times separately, then divide. If equal times are spent at two speeds, the arithmetic mean of the speeds works because each contributes equal time weight. If equal distances are covered, the slower segment consumes more time and has more influence on the overall speed. A twenty-kilometer trip at ten kilometers per hour followed by twenty kilometers at twenty kilometers per hour takes three hours, making the average forty-thirds kilometers per hour, below fifteen. The difference is a modeling issue, not a calculator precision issue.
A concentration compares component mass with total mixture mass. When pure water is added to saltwater, the salt mass stays fixed while the total grows. When pure salt is added, both numerator and denominator grow. To combine mixtures, compute the component amount in each, add those amounts, and divide by the combined total. An unweighted average of the two concentrations works only for equal mixture amounts. Before solving an equation, predict whether the concentration should rise or fall and between which values it must lie. That qualitative check catches a reversed numerator immediately.
Simple interest adds the same amount of interest each period because the principal used for interest stays fixed: interest = principal times rate times number of periods. Compound growth applies a multiplier each period, so the final amount after n periods is principal times (1 + rate) to the power n. If the rate is annual but time is given in months, convert time or the period consistently. Do not divide an effective annual compound rate by twelve unless the question specifies a nominal rate compounded monthly or otherwise permits that conversion. The wording determines the model.
A transport bill can combine a fixed fee with a distance charge. Suppose a discount applies only to the distance charge. Multiplying the entire bill by the discount factor would also reduce the fixed fee and produce the wrong model. Split the bill into its stated components, apply the change to the eligible component, and recombine them. This same reasoning applies to taxes on selected goods, tuition awards that exclude fees, and maintenance costs with a fixed annual charge.
Use a unit check at each stage. A price per kilometer multiplied by kilometers becomes money; adding a fixed amount of money is then meaningful. Adding a price per kilometer directly to a fixed fee would combine unlike quantities.
After solving, change one condition mentally. If the distance were zero, the bill should still equal the fixed fee. If the discount were zero, the original charge should return. Such boundary checks make a model reviewable without needing an official answer choice. The practice records your final numerical answer; use its explanation to diagnose whether any error came from the base, the units or the calculation.
A percent decrease is measured from the original, not the reduced value. A ratio of red to blue is not red divided by the total. Work times cannot be added as if they were work rates. Equal distances do not give equal time weights. In a mixture, adding material can change both the component amount and total amount.
Name the ratio quantities.
Copper and zinc have mass ratio 4:1 in a 75-gram alloy.
The ratio compares copper with zinc, not copper with total alloy.
Count the total parts.
4 + 1 = 5 equal mass parts.
Every part uses the same scale factor.
Find the scale factor.
75/5 = 15 grams per part.
The total mass fixes the size of one ratio part.
Compute and check copper mass.
4 times 15 = 60 grams.
The remaining fifteen grams is zinc, and 60:15 reduces to 4:1.
Represent the original price.
Let x be the original price in dollars.
The final price is 216 after a twenty percent reduction and an eight percent tax.
Apply the first multiplier.
Discounted price = 0.8x.
A reduction leaves eighty percent of the original.
Apply the second multiplier.
Tax-inclusive price = 1.08 times 0.8x.
The tax uses the discounted price as its base.
Solve the resulting equation.
0.864x = 216, so x = 250.
Divide by the combined multiplier rather than subtracting rates.
Check the two stages.
250 becomes 200, then 216.
Both stated changes lead back to the given final amount.
Identify the starting component.
A 90-gram solution is 20 percent salt, so it contains 18 grams of salt.
The remaining seventy-two grams is water.
Name the added amount.
Add x grams of pure salt.
The target concentration is forty percent.
Write the new numerator.
Salt mass = 18 + x.
All added material contributes to the salt amount.
Write the new denominator.
Total mass = 90 + x.
The same addition also increases total mass.
Solve the concentration equation.
(18 + x)/(90 + x) = 0.4 gives x = 30.
Multiplying out gives 18 + x = 36 + 0.4x.
Verify concentration and units.
48 grams of salt in 120 grams is forty percent.
Thirty grams is an added mass, not the final total or the final salt mass.
Convert times to rates.
Two workers finish a job alone in four and twelve hours: rates are 1/4 and 1/12 job per hour.
Assume their constant rates add without interference.
Combine the matching units.
1/4 + 1/12 = 1/3 job per hour.
Determine the time needed for one whole job.
Invert and check the rate.
A quantity increases from 80 to 92. What is the percent increase? Enter the percentage, not its decimal form.
The ratio of red to blue tiles is 3:5. There are 64 tiles in total. How many are blue?
A price of 150 dollars is reduced by 20 percent and then increased by 10 percent. What is the final price in dollars?
A discounted price is 68 dollars after a 15 percent reduction. What was the original price in dollars?
A solution contains 12 grams of salt in 80 grams of solution. How many grams of pure water must be added to make salt 10 percent of the new total mass?
A cyclist covers 24 kilometers at 12 kilometers per hour and returns along the same route at 8 kilometers per hour. What is the average speed for the whole trip in kilometers per hour?
One pump fills a tank in 6 hours and another fills it in 3 hours. Working together at constant rates, how many hours do they take to fill an empty tank?
A recipe uses flour and water in a mass ratio of 5:3. If the flour mass is 450 grams, what is the total mass of flour and water in grams?
A fund earns 4 percent annual simple interest on a principal of 2500 dollars for 3 years. How many dollars of interest are earned?
A machine produces 18 units every 12 minutes at a constant rate. How many units does it produce in 50 minutes?
A transport service charges 6 dollars plus 1.5 dollars per kilometer. A promotion reduces only the distance charge by 20 percent. What is the promoted total charge in dollars for a 15-kilometer trip?
Answer three reserved questions without hints. These cases are separate from lesson practice.
Two printers process 8 and 12 pages per minute respectively. Together at constant rates, how many minutes do they need to process 360 pages?
A 120-gram mixture is 25 percent sugar by mass. How many grams of sugar must be added to make the resulting mixture 40 percent sugar?
A membership rises from 240 to 276 people. What is the percent increase? Enter the percentage.
Tomorrow, explain why equal-distance average speed is not the average of the two speeds. Then solve a reverse-discount question and a mixture question without consulting the examples. Review Algebra I units if your equation adds unlike quantities.
17. Finish a work-rate model, step 3
The joint time is three hours.
This is shorter than either individual completion time.