🇨🇩 Algebra: integers, decimals, fractions

Decimals and fractions

⏱ 25 minOfficial programme: CD-E7-ALG-02

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🎯 In this lesson, you will learn to

Operate on decimals; solve commerce/environment problems.
Simplify, compare and operate on fractions.

The lesson

Introduction

At the Matadi market, Furaha buys $2.5$ kg of sugar and $1.5$ l of palm oil. Then she uses $\frac{2}{5}$ of a water can for cooking. To pay the right amount, give change or share a quantity, you need to compute with decimal numbers and fractions. That is what this lesson is about.

Explanation

Decimal numbers

A decimal number has a finite number of digits after the decimal point: $3.75=\dfrac{375}{100}$ . The set of decimals is written $\mathbb{D}$ .

Addition, subtraction: line up the decimal points (you may add zeros: $6.2=6.20$ ).
Multiplication: multiply without the decimal points; the product has as many digits after the point as the two factors together: $0.6\times0.4=0.24$ .
Dividing by a decimal: multiply the dividend and the divisor by the same power of $10$ to make the divisor whole: $7.5\div0.25=750\div25=30$ .
Multiplying by $10$ , $100$ , $1\,000$ moves the decimal point $1$ , $2$ , $3$ places to the right.

Explanation

Fractions: simplifying and comparing

A fraction $\dfrac{a}{b}$ ( $b\ne0$ ) has a numerator $a$ and a denominator $b$ .

A fraction does not change when both terms are multiplied or divided by the same non-zero number: $\dfrac{3}{4}=\dfrac{6}{8}$ .
Simplifying: divide both terms by a common divisor. With the greatest common divisor, you get the irreducible fraction directly: $\dfrac{36}{48}=\dfrac{36\div12}{48\div12}=\dfrac{3}{4}$ .
Comparing: with the same denominator, the greater fraction has the greater numerator. Otherwise, first rewrite them with the same denominator.

Explanation

Operations on fractions

Sum and difference: the same denominator is required: $\dfrac{a}{d}+\dfrac{b}{d}=\dfrac{a+b}{d}$ . Otherwise, rewrite them with the same denominator.
Product: $\dfrac{a}{b}\times\dfrac{c}{d}=\dfrac{a\times c}{b\times d}$ .
Quotient: dividing by $\dfrac{c}{d}$ means multiplying by its reciprocal $\dfrac{d}{c}$ ( $c\ne0$ ): $\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\times\dfrac{d}{c}$ .
Fraction of a quantity: $\dfrac{2}{5}$ of $20$ l is $\dfrac{2\times20}{5}=8$ l.

The order of operations stays the same: products and quotients before sums and differences. Simplify the final result.

Worked example

Problem: Furaha buys $2.5$ kg of sugar at $4\,200$ FC per kilogram and $1.5$ l of palm oil at $3\,800$ FC per litre. She pays with a $20\,000$ FC note. How much change does she get?

Step 1: Sugar: $2.5\times4\,200=10\,500$ FC.

Step 2: Oil: $1.5\times3\,800=5\,700$ FC.

Step 3: Total spent: $10\,500+5\,700=16\,200$ FC.

Step 4: Change: $20\,000-16\,200=3\,800$ FC.

Answer: Furaha gets $3\,800$ FC.

Worked example

Problem: Make $\dfrac{36}{48}$ irreducible, then compare it with $\dfrac{5}{8}$ .

Step 1: Common divisors of $36$ and $48$ : the greatest is $12$ . So $\dfrac{36}{48}=\dfrac{3}{4}$ .

Step 2: Same denominator: $\dfrac{3}{4}=\dfrac{6}{8}$ .

Step 3: $6>5$ , so $\dfrac{6}{8}>\dfrac{5}{8}$ .

Answer: $\dfrac{36}{48}=\dfrac{3}{4}$ and $\dfrac{3}{4}>\dfrac{5}{8}$ .

Worked example

Problem: Compute $E=\dfrac{2}{3}+\dfrac{5}{6}\times\dfrac{3}{10}$ .

Step 1: The product first: $\dfrac{5}{6}\times\dfrac{3}{10}=\dfrac{15}{60}=\dfrac{1}{4}$ .

Step 2: Same denominator $12$ : $\dfrac{2}{3}=\dfrac{8}{12}$ and $\dfrac{1}{4}=\dfrac{3}{12}$ .

Step 3: $\dfrac{8}{12}+\dfrac{3}{12}=\dfrac{11}{12}$ .

Answer: $E=\dfrac{11}{12}$ .

Watch out

Watch out: never add the denominators. $\dfrac{1}{2}+\dfrac{1}{3}$ is not $\dfrac{2}{5}$ : rewrite with the same denominator, $\dfrac{3}{6}+\dfrac{2}{6}=\dfrac{5}{6}$ . For decimals, count the digits after the point carefully in a product: $0.3\times0.2=0.06$ , not $0.6$ .

Key points

Decimals: line up the points for $+$ and $-$ ; count the digits after the point for $\times$ .
To divide by a decimal, make the divisor whole by multiplying both numbers by $10$ , $100$ …
Irreducible fraction: divide both terms by their greatest common divisor.
Sum of fractions: same denominator first; product: numerators together, denominators together.
Dividing by a fraction means multiplying by its reciprocal.

✍️ Exercises

Answer one exercise at a time. Your answer is checked right away, with an explanation.

Exercise 1 of 8

★ Easy

Compute $12.5+3.75-6.2$ .

🎯 In this lesson, you will learn to

The lesson

👋 Introduction

📘 Explanation

📘 Explanation

📘 Explanation

✏️ Worked example

✏️ Worked example

✏️ Worked example

⚠️ Watch out

⭐ Key points

✍️ Exercises

Introduction

Explanation

Explanation

Explanation

Worked example

Worked example

Worked example

Watch out

Key points