Вопрос школьника по предмету Математика
из подготовительных ЕГЭ по математике (база!)
В обменном пункте можно совершить одну из двух операций:
• за 2 золотые монеты получить 3 серебряные и одну медную;
• за 7 серебряных монет получить 3 золотые и одну медную.
У Николая были только серебряные монеты. После обменного пункта золотых монет у него не появилось, зато появилось 20 медных. На сколько уменьшилось количество серебряных монет у Николая?
Ответ учителя по предмету Математика
В результате обмена можно было совершить такие действие:
— за
2 золотые монеты получить 3 серебряные и 1 медную:
2з=3с+1м
—
за 7 серебряных монет получить 3 золотые и 1 медную:
7с=3з+1м
1. У Николая были только серебряные монеты, значит первой операцией было:
7с=3з+1м. У него появилось 3 золотые монеты и 1 медная, а серебряные уменьшились на 7.
2. Затем Николай обменял золотые монеты (т.к. после обменного пункта золотых монет у него не появилось):
2з=3с+1м. У него осталась одна золотая монета, прибавилось 3 серебряных и 2 медных.
Чтобы у Николая появились 20 медных монет, ему надо было совершить 20 таких операций.
Составим систему уравнений:
{2з=3с+1м
{7с=3з+1м
Решим систему методом сложения:
{2з=3с+1м | *3
{-3з=1м-7с | *2
{6з=3м+9с
{-6з=2м-14с
(6з+(-6з))+(9с-14с)+(3м+2м)=0-5с+5м=5м-5с
В конце обмена У Николая появилось 20 медных монет, значит количество серебряных монет должно уменьшиться в 20м:5м=4 раза
5с=5*4=20 серебряных монет
Ответ: Николай потерял 20 серебряных монет.
или
всего было совершено 20 операций (в конце осталось 20 медных монет):
— чтобы в конце обмена не осталось золотых монет, Николай на каждые две вторые операции (7с=3з+1м, 3з*2=6золотых монет) совершал три первые операции (2з=3с+1м; 2з*3=6з), дающие в итоге 6з-6з=0 золотых монет.
— из 20 операций Николай провел 12 первых и 8 вторых операций, чтобы золотых получилось по итогу 0.
В конце обмена количество серебряных монет составит:
+3с*х(количество первых операций) — 7*у (количество вторых операций)
х=12
у=8
3с*12-7с*8=36с-56с=-20 серебряных монет потратил Николай
ОТВЕТ: Николай потратил 20 серебряных монет.
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