Procenty i proporcje • lekcja 1 z 9
Procenty podstawowe i sens procentu
W dwóch klasach połowa uczniów wybiera wycieczkę rowerową. Czy w obu jedzie tyle samo osób? Procent opisuje udział, a nie gotową liczbę osób. Nauczysz się odczytywać ten udział i wskazywać jego podstawę.
Po tej lekcji: wyjaśnisz sens procentu, zamienisz procenty na ułamki i odwrotnie oraz wskażesz, co oznacza 100%.Na początek: Przypomnij sobie ułamki zwykłe i dziesiętne. Zapis 1/4 oznacza jedną z czterech równych części.
Sto równych części — także w małej grupie
W klasie 8a jest 20 uczniów, a w 8b — 28. Połowa pierwszej klasy to 10 osób, połowa drugiej to 14. W obu przypadkach udział wynosi 50%, chociaż liczby osób są różne. Procenty pozwalają opisać, jak duża jest część w porównaniu z wybraną całością.
Definicja.Jeden procent, czyli 1%, to jedna setna: 1% = 1/100 = 0,01. Cała wielkość przyjęta za podstawę odpowiada 100%. Liczba procentów mówi, ile setnych tej podstawy rozważamy.
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Jedna kratka stanowi 1% kwadratu, jeden rząd — 10%. Zaznaczone trzy pełne rzędy to 30%, czyli 30/100 = 3/10 = 0,3. Siatka ma sto pól, ale zbiór opisywany procentowo nie musi mieć stu elementów: procenty opisują również klasy dwudziestoosobowe.
Jak przejść między zapisami?
Przykład: 35% i trzy czwarte
- Aby zamienić 35% na ułamek, zapisz 35 setnych: 35/100.
- Skróć ułamek przez 5: 35/100 = 7/20. W zapisie dziesiętnym jest to 0,35.
- W przeciwną stronę: 3/4 rozszerzamy przez 25. Otrzymujemy 75/100, czyli 75%.
Kontrola: 35% jest mniejsze od połowy, a 75% większe od połowy. Ułamki 0,35 i 0,75 zachowują te same relacje.
Gdy ułamek dziesiętny jest już znany, policz liczbę setnych: 0,08 = 8/100 = 8%. Dla 0,125 otrzymasz 12,5%, ponieważ 0,125 · 100 = 12,5. Aby przejść od procentu do liczby, dzielisz liczbę procentów przez 100. Aby zapisać liczbę procentowo, mnożysz ją przez 100 i dopisujesz znak %.
Mały przecinek, duża różnica
0,5 = 50%, natomiast 0,5% = 0,005. Pół procentu to połowa jednego procentu, a nie połowa całości. Podobnie 5% = 0,05, nie 0,5. Zawsze sprawdź, czy w danych występuje znak procentu.
Czy można przekroczyć 100%?
Tak, gdy porównujemy wielkość z mniejszą podstawą. Planowano przejść 10 km, a przebyto 15 km: to półtora planu, czyli 150% planu. Natomiast uczniowie jednej klasy noszący okulary nie mogą stanowić 150% tej klasy. Udział części w dodatniej całości mieści się między 0% a 100%. 0% oznacza brak tej części, a 100% — całą podstawę.
Teraz sprawdź swoje rozumienie
Zapisz własne rozwiązanie przed odsłonięciem odpowiedzi. Jeśli nie wiesz, jak zacząć, skorzystaj ze wskazówki. Po sprawdzeniu wyjaśnij własnymi słowami, dlaczego wybrane działanie pasuje do zadania.
Zadanie 1
Wskazówka
Zacznij od 45/100.
Rozwiązanie i kontrola
45/100 = 9/20 = 0,45. Skracamy przez 5; oba zapisy oznaczają mniej niż połowę.
Zadanie 2
Wskazówka
Wyraź każdą liczbę w setnych.
Rozwiązanie i kontrola
0,07 = 7%; 3/5 = 60/100 = 60%; 1,25 = 125%. Ostatni udział przekracza jedną podstawę.
Zadanie 3
Wskazówka
Podziel 0,8 przez 100.
Rozwiązanie i kontrola
0,8% = 0,008. Jest to mniej niż 0,01 = 1%.
Zadanie 4
Wskazówka
Połowa każdej klasy jest liczona osobno.
Rozwiązanie i kontrola
Podstawami są odpowiednio 24 i 30 osób. Trenuje 12 i 15 osób. Równy procent nie oznacza równej liczby osób.
Zadanie 5
Wskazówka
150% to jedna całość i jeszcze jej połowa.
Rozwiązanie i kontrola
20 + 10 = 30 zadań. Nie chodzi o 150 zadań, lecz o półtora ustalonego planu.
Najważniejsze zależności
Najpierw nazwij podstawę, czyli 100%. Procent to zapis liczby w setnych. Ten sam procent różnych podstaw może oznaczać różne wartości.
