MATHHX B
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5.1 Procentregning
Procent betyder "hundrededele". Spiser man f.eks. \(8\%\) af en kage betyder det altså at hvis kagen er skåret i \(100\) stykker, har man spist \(8\) stykker. Har man en pris der stiger med \(10\%\) betyder det at hvis vi deler
prisen op i \(100\) dele, skal vi lægge \(10\) af disse oven i den oprindelige pris osv.
Der findes flere metoder til at regne procentregningsopgaver. Kan man allerede en metode, kan man vægle at holde fast i den og springe direkte til øvelserne i dette afsnit. Dog skal man være opmærksom på, at den metode, man
allerede kender, muligvis ikke er tilstrækkelig til at regne alle øvelserne. I det følgende skal vi se på to skemaer, som viser to typer af metoder til at regne procentregningsopgaver. Metoderne vist i det første skema minder om dem,
man typisk bliver undervist i i folkeskolen.
.
Spørgsmål
Svar
Hvad er \(10\%\) af \(500\)?
\(\frac {500}{100}\cdot 10=50\)
\(10\%\) af et tal er \(500\). Hvor stort er det oprindelige tal?
\(\frac {500}{10}\cdot 100=5000\)
Hvor mange procent udgør \(50\) ud af \(500\)?
\(\frac {50}{500}\cdot 100\%=10\%\)
Læg \(10\%\) til \(500\).
\(500+\frac {10}{100}\cdot 500=550\)
Træk \(10\%\) fra \(500\).
\(500-\frac {10}{100}\cdot 500=450\)
Et beløb vokser fra \(500\) kr. til \(600\) kr. Hvor mange procent er det vokset?
\(\frac {600-500}{500}\cdot 100\%=20\%\)
Et beløb er faldet fra \(600\) kr. til \(500\) kr. Hvor mange procent er det faldet?
\(\frac {600-500}{600}\cdot 100\%=16{,}67\%\)
Efter vi har lagt \(10\%\) til et beløb har vi \(500\) kr. Hvad havde vi før vi lagde \(10\%\) til?
\(\frac {500}{100+10}\cdot 100=454{,}55\)
Efter vi har fjernet \(10\%\) fra et beløb har vi \(500\) kr. Hvad havde vi oprindeligt?
\(\frac {500}{100-10}\cdot 100=555{,}56\)
Skema 1
Metoderne i skema 1 er simple at bruge, da man blot erstatter tallene i eksemplerne med sine egne. Har man en god forståelse af hvad procent betyder, er der dog hurtigere metoder til at udregne procenter. Disse er vist i skema 2:
.
Spørgsmål
Svar
Hvad er \(10\%\) af \(500\)?
\(0{,}1\cdot 500=50\)
\(10\%\) af et tal er \(500\). Hvor stort er det oprindelige tal?
\(\frac {500}{0{,}1}=5000\)
Hvor mange procent udgør 50 ud af \(500\)?
\(\frac {50}{500}=0{,}1=10\%\)
Læg \(10\%\) til \(500\).
\(1{,}1\cdot 500=550\)
Træk \(10\%\) fra \(500\).
\(500\cdot 0{,}9=450\)
Et beløb vokser fra \(500\) kr. til 600 kr. Hvor mange procent er det vokset?
\(\frac {600}{500}-1=0{,}2=20\%\)
Et beløb er faldet fra 600 kr. til \(500\) kr. Hvor mange procent er det faldet?
\(1-\frac {500}{600}=0{,}1667=16{,}67\%\)
Efter vi har lagt \(10\%\) til et beløb har vi \(500\) kr. Hvad havde vi før vi lagde \(10\%\) til?
\(\frac {500}{1{,}1}=454{,}55\)
Efter vi har fjernet \(10\%\) fra et beløb har vi \(500\) kr. Hvad havde vi oprindeligt?
\(\frac {500}{0{,}9}=555{,}56\)
Skema 2
Øvelse 5.1.1
Du skal regne følgende procentregningsopgaver. Du bestemmer selv metoden, men jeg anbefaler metoderne for Skema 2.
a) En elev drikker \(150\) ml af en \(500\) ml Faxe Kondi. Hvor mange procent af sodavanden har eleven drukket?
b) En elev har \(10\%\) fravær i matematik. Der har været 20 matematikmoduler. I hvor mange moduler har eleven været fraværende?
c) En elev køber en telefon med sin far. De aftaler at eleven skal betal \(30\%\) af prisen. Eleven ender med at betale 2700 kr. Hvad kostede telefonen?
d) En vare koster \(200\) kr. og stiger derefter med \(20\%\). Hvad koster varen efter prisstigningen.
e) Momssatsen i England er \(20\%\) (i hvert fald på tidspunkt opgaven er skrevet). En vare koster 30 GBP med engelsk moms, hvad koster den uden?
f) Brøndby IF tilbød i 2013/2014 sæsonkort for \(995\) kr. Var man aktionær kunne man får det \(20\%\) billigere. Hvad kostede et sæsonkort for
aktionærer?
g) En elev køber et par sko til \(840\) kr. Skoene var sat \(30\%\) ned. Hvad var normalprisen?
h) En elev arbejder i en butik der sælger slik. En kunde køber en pose piratos for \(10\) kr. som normalt koster \(20\) kr. Hvor mange procent har
kunden fået i rabat?
i) Priserne på hindbærsnitter stiger fra \(10\) til \(12\) kr. Hvor stor er prisstigningen i procent?
Løsning 5.1.1
a) \(30\%\)
b) \(2\)
c) \(9.000\) kr.
d) \(240 kr.\)
e) \(25\) GBP
f) \(796\) kr.
g) \(1200 kr.\)
h) \(50\%\)
i) \(20\%\)
Forklaring af metoderne
En ting er at kunne regne procentregningsopgaver, noget andet er at kunne forklare metoderne.
Vi vil nu se på den første metode i skema 1. Det påstås altså at vi kan finde \(10\%\) af \(500\) ved at regne:
\[\frac {500}{100}\cdot 10=50.\]
Hvorfor er det mon rigtigt? Jo altså, procent betyder hundrededele, så ved at dividere \(500\) med \(100\) finder vi \(1\) hundrededel. Dem skal vi så have \(10\) af, og derfor ganger vi med \(10\). Hermed er den første metode i
skema 1 forklaret.
Øvelse 5.1.2
Nu er det din tur til at forklare.
Løsning 5.1.2
a) Spørg mig.
b) Spørg mig.