U3. Proportionality

1. Introductory activities

Wayground (former Quizizz)

2. Directly proportional magnitudes

Something you should remember:

Ratio (razón): A way to compare two numbers, the division of both numbers. 4:3 "ratio four to three", as in the scale of a map, 1:50000

Proportion (proporción): two equal ratios 4:3 = 1600:1200, usually expressed as fractions.

Directly proportional magnitudes: one rise as the other rise, their corresponding values form a proportion. The ratio is called, in this context, constant of proportionality. 

Examples of different magnitudes (not all are proportional):

• Price in euros and kilograms of oranges.

• Time in minutes to do a task and people doing it.

• Shoe size and height of people in cm.

• Weight of a jar with marbles and number of marbles in it.

Exercise: Types of Proportionality

For each pair of quantities below, decide whether they are: directly proportional, Inversely proportional, or not proportional. Explain your choice briefly in each case.

A. The number of hours you work — and the amount of money you earn (at a fixed hourly rate).

B. The speed of a car — and the time it takes to travel a fixed distance.

C. The number of slices you cut a pizza into — and the size of each slice.

D. The number of students in a class — and the average grade of the class.

E. The length of one side of a square — and its area.

F. The distance you walk — and the time you spend walking (at a constant speed).

G. The price of one notebook — and the total price of 5 identical notebooks.

H. The temperature in °C — and the temperature in °F.

Can you invent two more pairs of quantities of every type?

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Algorithm on solving many word problems on proportionality

As in: Antonio buy 4 workbooks by 6 euros. Paula is going to buy 3 workbooks. How much will they cost? Rubén has 15 euros. How many workbooks can he buy?

1. Use a table with two rows for the magnitudes. 

2. Wonder if the magnitudes are directly proportional or not (double the first goes with double the second).

3. Express the relations in the problem as a proportion. 

4. Solve for x (cross products must be the same).

Solve the following easy problems using that method:

1. If 4 hours of work earn you €36, how much will you earn if you work 7 hours at the same rate?

2. A car travels 180 km in 3 hours. How far will it travel in 5 hours at the same speed?

3. 5 notebooks cost €12.50. How much will 8 notebooks cost?

4. To make pancakes for 4 people, you need 200 g of flour. How much flour is needed for 10 people?

5. A car uses 8 liters of petrol to travel 120 km. How many liters are needed for 300 km?

6. Downloading 15 songs takes 6 minutes. How long will it take to download 40 songs at the same rate?

7. A painter covers 12 m² in 30 minutes. How many square meters can he paint in 2 hours?

8. A phone company charges €9 for 3 GB of data. How much would 10 GB cost?

9. If you save €45 in 3 weeks, how much will you save in 8 weeks at the same rate?

10. A printer produces 120 pages in 8 minutes. How many pages will it print in 15 minutes?

Solutions (there is one wrong): €63, 300 km, €20, 500 g, 20 L, 16 min, 48 m², €30, €130, 225 pages

4. Inverse proportion word problems

1. What does “inverse” mean?

“Inverse” means opposite.
In mathematics, two quantities are inversely proportional when one increases while the other decreases, in such a way that their product stays the same.

x · y = constant

That means:

• If one doubles, the other becomes half.

• If one becomes three times bigger, the other becomes three times smaller.

2. Everyday examples

We can find inverse proportion in real life:

• Speed and travel time: The faster you go, the less time you need to travel the same distance.
  Example: If you ride your bike twice as fast, the time to reach school is divided by two.

• Number of people and amount of work per person:
  If more people help with a task, each one does less work.
  Example: If 2 friends clean a classroom in 1 hour, 4 friends will take only 30 minutes if they work at the same rate.

3. Inverse sharing problems

Sometimes we need to divide an amount of money, points, or rewards inversely proportional to some numbers.
That means: those with larger numbers get smaller parts, and those with smaller numbers get larger parts.

We use the reciprocal of the given numbers (1 over the number).

We'll see it below.

4. Solving inverse prorportion word problems

a) Set the magnitudes in a table, x, y.

b) Identify that the relationship is inverse (one goes up, the other goes down).

c) Use the constant product rule: x1· y1 = x2 · y2

Substitute the known values and find the missing one.

Solve the following easy problems:

1. Six workers can build a wall in 10 days. How long will it take 15 workers to build the same wall?

2. A car takes 5 hours to travel a certain distance at 80 km/h. How long would it take if it goes at 100 km/h?

3. Four machines can fill 200 bottles in 3 hours. How long would it take six machines to fill the same number of bottles?

4. 12 students can clean the school yard in 45 minutes. How long will it take 9 students to clean it?

5. One pipe can fill a swimming pool in 6 hours. How long will it take three pipes working together?

5. Taller de calculadoras

Taller de calculadoras.

Practica con los siguientes problemas:

1: Lucía y María compran juntas un boleto de lotería de 20 €. Lucía pone 15 € y María 5 €. Ganan un premio de 400 €. ¿Cuánto le corresponde a cada una para que el reparto sea justo?

2: Tres amigos (Sofía, Marc y Alejandro) crean un canal conjunto de noticias sobre videojuegos y música. Este mes han conseguido sus primeros 450 € por patrocinio y deciden repartirlos de forma directamente proporcional al número de vídeos editados por cada uno: Sofía editó 12 vídeos, Marc 8 vídeos y Alejandro 5 vídeos. ¿Cuánto dinero le corresponde a cada uno?

3: Para financiar la excursión de fin de curso, el AMPA premia a los tres alumnos que más boletos han vendido repartiendo un vale de 150 € para material deportivo o tecnológico, de manera directamente proporcional a los talonarios vendidos. Daniela vendió 12 talonarios, Iker 10 talonarios y Youssef 8 talonarios. ¿Qué cantidad de vale le corresponde a cada uno?

4: Tres amigos alquilan una casa rural el fin de semana por un total de 360 €. Se reparten el coste en proporción a las noches que se queda cada uno: 3 noches, 2 noches y 1 noche.

5: En el torneo de eSports del instituto, la organización entrega una tarjeta de regalo de 120 € a repartir entre los tres primeros clasificados de forma directamente proporcional al número de partidas ganadas en la fase final: Paula logró 11 victorias, Hugo 9 victorias y Mateo 4 victorias. ¿Cuánto dinero en la tarjeta recibe cada uno?

7. Percentages

Increasing and decreasing percentajes as products

Increasing a quantity, for example 23 by 5% means that you will add 5% to the initial 100%. The final result will be 105% of the initial quantity. So you can calculate that amount by multiplying 23 · 105% = 23 · 1.05 = 24.15.

For the same reason, decreasing, let's say, 39 by 12% can be calculated as 88% of 39, since 88% = 100% - 12%. Doing the maths, 39 · 88% = 39 · 0.88 = 34.32

1. Increase each of the following quantities by the given percentage, usign the multiplication method: 

   • 60 € by 4%; 

   • 12 kg by 8%; 

   • 450 g by 5%; 

   • 545 m by 10% 

2. Decrease each of the following quantities by the given percentage, using the multiplication method:

   • 34 € by 12%; 

   • 75 € by 20%; 

   • 340 kg by 15%; 

   • 670 cm by 23%

3. A phone costs €320. During a sale, the price is reduced by 25%. What is the new price?

4. A pair of trainers cost €80. The shop increases all prices by 10%. How much do they cost now?

5. A jacket that used to cost €120 is now €96. What percentage discount was applied?

6. A concert ticket costs €50. If VAT (21%) is added, what is the final price including tax?

7. A bicycle was sold for €270 after a 10% discount. What was its original price?

8. A computer is first discounted by 20%, and then an extra 10% is applied. What is the total percentage discount?

9. A product’s price is increased by 15% in January and then by another 10% in March. What is the final percentage increase overall?

10. A T-shirt originally costs €40. It is first reduced by 25%, and later, the new price increases by 20%. What is the final price?

11. The population of a town increases by 5% in one year and decreases by 8% the next year. What is the overall percentage change?

12. 35% of the students in a school are boys. If there are 420 students in total, how many are girls?

13. A student got 72 marks in a test. If this represents 80% of the total, what was the maximum mark?

14. A shop sells a hoodie for €45, which includes a 10% profit. What was the cost price for the shop?

15. A laptop’s value after a 20% depreciation is €640. What was its original value?

16. After a 15% increase, the price of a backpack is €57.50. What was its original price?

17. A phone’s price was €300, but after a discount, it was sold for €255. What percentage discount was applied?

Solutions (one is incorrect): 

8. Simple Interest

1. What is interest?

When we save money in a bank, the bank gives us some extra money as a reward. When we borrow money, we must pay back more than we borrowed. That extra money is called interest.

Interest depends on:

• The initial amount of money (called principal, ( P ))

• The interest rate (a percentage, ( r ))

• The time the money is saved or borrowed (( t ))

Interest connects directly with percentages and proportionality, because it increases the amount in a proportional way.

2. What is simple interest?

In simple interest, the interest is always calculated on the initial amount only.

Formula:

I = P · r · t

 where:

• ( I ) = interest earned

• ( P ) = principal (the starting amount)

• ( r ) = annual interest rate (in decimal form, e.g. 5% → 0.05)

• ( t ) = time in years

The total amount after the time is:
C = P + I

3. Example: Simple Interest

Emma saves 500 € in a bank for 3 years at an interest rate of 4% per year.

I = 500 · 0.04 · 3 = 60

So, the total amount after 3 years is:

A = 500 + 60 = 560 €
Explanation:
Every year, Emma earns 4% of 500 = 20 €.
After 3 years, 3 · 20 € = 60 €.
The increase is directly proportional to the time and the rate.

4. Practice: Simple Interest

• You invest 300 € for 4 years at 5% per year. How much interest will you earn? What will be the total amount?

• A shop offers a 3% monthly interest on unpaid bills. If you owe 200 € for 2 months, how much will you pay in total?

• The interest you get in simple interest is directly proportional to which quantities?

5. What is compound interest?

In compound interest, the interest is added to the total every period (usually each year).
Then, in the next period, the new total is used to calculate interest again.
That means you earn interest on interest.

Formula:

A = P·(1+r)t 

where:

• ( A ) = total amount after ( t ) years

• ( P ) = principal

• ( r ) = annual interest rate (as a decimal)

• ( t ) = time in years

The interest is (I = A - P).

6. Example: Compound Interest

Emma saves 500 € for 3 years at 4% per year, with compound interest.

[ A = 500 · (1 + 0.04)3 = 500 · 1.124864 = 562.43 € ]

So, Emma earns 62.43 € in total interest.

Explanation:
Each year the interest increases slightly because it is calculated on the new total, not just the initial 500 €.

Practice: Compound Interest

1. You invest 400 € for 2 years at 5% interest per year. Find the total amount and the interest earned.

2. A company borrows 1 000 € at 10% per year for 3 years. How much will they pay back in total? Compare the results of simple and compound interest. Which grows faster? Why?

3. Imagine you deposit 200 € in a bank that offers 3% compound interest per year. Calculate the total amount after 1, 2, 3, 4 and 5 years.

9. Mental calculation: percentajes

Bingo de porcentajes.

10. COMPOUNDO PROPORTIONALITY & INVERSE SHARING

🐱 Warm-Up Riddle: The Cats & Mice Paradox

Before diving into complex problems, test your intuition with this classic mathematical riddle:

The Riddle:
If 3 cats catch 3 mice in 3 hours, how many cats are needed to catch 100 mice in 100 hours?

⚙️ Compound Proportionality (3 Variables)

When a problem involves three or more quantities (such as workers, days, hours per day, or machines and production), we are dealing with compound proportion.

How to Solve Compound Proportionality Step-by-Step:

1. Organize data in a table: Place the unknown variable (x) in the last column.

2. Compare each variable with the target variable (x):

   • Directly Proportional (D): If increasing one quantity increases the target quantity.

   • Inversely Proportional (I): If increasing one quantity decreases the target quantity.

3. Set up the proportion equation: (Initial x) / (New x) = (Ratio 1) · (Ratio 2)
   (Remember to invert the fraction for any Inversely Proportional relationship!)

📝 Worked Example: Printing School Components

Problem: 5 industrial 3D printers operating 8 hours a day produce 1,000 components in 6 days. How many days will it take 6 printers operating 10 hours a day to produce 1,500 components?

• Data Table:
  Printers: 5 → 6
  Hours/day: 8 → 10
  Components: 1,000 → 1,500
  Days: 6 → x

• Analyzing Relationships:
  Printers vs. Days: More printers → Fewer days (INVERSE - I)
  Hours/day vs. Days: More hours per day → Fewer days (INVERSE - I)
  Components vs. Days: More components → More days (DIRECT - D)

• Setting up the Equation: 6 / x = (6 / 5) · (10 / 8) · (1,000 / 1,500)
  6 / x = (6 · 10 · 1,000) / (5 · 8 · 1,500)
  6 / x = 60,000 / 60,000 = 1
  x = 6 / 1 = 6 days
  
  

Answer: It will take 6 days.

🎮 Inverse Proportional Sharing (Inverse Distribution)

In an inverse proportional sharing, the rule is: the greater your quantity, the smaller your share.
Example: Distributing a gaming prize where players lose money for penalties, or sharing a cost based on days absent.

The Reciprocal Method Rule: To share a quantity N inversely proportional to a, b, c:

1. Take the reciprocals of the numbers: 1/a, 1/b, 1/c.

2. Find a common denominator for the fractions.

3. Perform a Direct Proportional Sharing using the numerators!

🏆 Worked Example: eSports Tournament Prize

Problem: Three friends (Alex, Beatriz, and Charlie) win a total prize pool of 180 € in a VR Gaming Tournament. The organizer decides to distribute the prize inversely proportional to the number of penalties/lives lost during the finals:

• Alex lost 2 lives

• Beatriz lost 3 lives

• Charlie lost 6 lives

How much money does each player receive?

Step-by-Step Solution:

1. Write down the reciprocals of lives lost:

   • Alex: 1/2

   • Beatriz: 1/3

   • Charlie: 1/6

2. Convert to a common denominator (LCM of 2, 3, 6 = 6):

   • Alex: 3/6

   • Beatriz: 2/6

   • Charlie: 1/6

3. Use the numerators (3, 2, 1) for direct sharing:

   • Total parts: 3 + 2 + 1 = 6 parts

   • Value per part (k): k = 180 € / 6 parts = 30 € per part

4. Calculate final shares:

   • Alex (3 parts): 3 · 30 € = 90 €

   • Beatriz (2 parts): 2 · 30 € = 60 €

   • Charlie (1 part): 1 · 30 € = 30 €

(Check: 90 + 60 + 30 = 180 €. Fewer lives lost = More prize money!)

🎯 Practice Challenge!

Try solving these two problems in your notebook before checking the answers:

1. The Construction Challenge: 8 builders take 12 days working 9 hours a day to build a wall. How many days will 6 builders working 8 hours a day take to build the exact same wall?

2. The Library Fine Share: Two students share a 33 € book replacement fee inversely proportional to the number of days they returned books late (Student A was 2 days late, Student B was 4 days late). How much does each pay?

11. Situación de aprendizaje: elecciones

Elecciones.

More Word problems

1. A car travels 100 km with 4.5 litres of fuel. How far will it travel with 7 litres?

2. It take 8 students 9 hours to make a class work. How long would it take 6 students to make the same work?

3. It takes 12 people 15 days to harvest a crop of raspberries. How long would it take 18 people?

4. £1 is worth 1.21 €. How many euros is £9 worth?

5. A motorist drives at a steady speed and goes 232 km in 3 hours:

   1. What distance will she cover in 5 hours?

   2. How long will it take her to cover 193 km?

6. 5 litres of paint covers an area of 60 m². 

   1. What area will 15 litres of paint cover?

   2. How much paint is needed to cover an area of 240 m²?

7. A field provides grazing for 18 sheep for 8 days. How many days grazing would it provide for 24 sheep?

8. It takes an hour to mow a lawn using a mower with blades 14 inches wide. How long would it take using a mower with blades 12 inches wide?

9. 3 cm³ of aluminium weigh 8.1 g. Work out the weight of 10 cm³.

Criterios de evaluación y saberes básicos implicados en la unidad

Criterios de Evaluación

MAT.1.1.1.Iniciarse en la interpretación de problemas matemáticos sencillos, reconociendo los datos dados, estableciendo, de manera básica, las relaciones entre ellos y comprendiendo las preguntas formuladas.

MAT.1.1.2. Aplicar, en problemas de contextos cercanos de la vida cotidiana, herramientas y estrategias apropiadas, como pueden ser la descomposición en problemas más sencillos, el tanteo, el ensayo y error o la búsqueda de patrones, que contribuyan a la  resolución de problemas de su entorno más cercano.

MAT.1.1.3.Obtener las soluciones matemáticas en problemas de contextos cercanos de la vida cotidiana, activando los conocimientos necesarios, aceptando el error como parte del proceso.

MAT.1.2.1. Comprobar, de forma razonada la corrección de las soluciones de un problema, usando herramientas digitales como calculadoras, hojas de cálculo o programas específicos.

MAT.1.2.2. Comprobar, mediante la lectura comprensiva, la validez de las soluciones obtenidas en un problema comprobando su coherencia en el contexto planteado y evaluando el alcance y repercusión de estas soluciones desde diferentes perspectivas: igualdad de género, sostenibilidad, consumo responsable, equidad o no discriminación.

Saberes Básicos

MAT.3.A.3.1, MAT.3.B.1.2, MAT.3.D.4.2, MAT.3.E.2.3, MAT.3.A.1.1, MAT.3.A.4.4, MAT.3.D.6.2, MAT.3.D.6.3, MAT.3.C.4.1, MAT.3.D.1.1, MAT.3.D.2.1, MAT.3.A.2.4, MAT.3.A.4.2, MAT.3.E.1.2, MAT.3.E.1.3

Preguntas del examen

Problema 1. Problema de proporcionalidad directa.

Problema 2. Problema de repartos.

Problema 3. Problema de porcentajes.

Problema 4. Problema de interés compuesto.

Problema 5. Problema de proporcionalidad inversa.